Coordinate Geometry-Vocabulary List
Coordinate Geometry Vocabulary
- Cartesian Plane: A plane defined by two perpendicular number lines, called the x-axis and y-axis, intersecting at a point called the origin.
-
Origin: The point where the x-axis and y-axis intersect, represented as
. - Quadrants: The four regions of the Cartesian plane, divided by the x-axis and y-axis. They are labeled as Quadrant I, II, III, and IV.
-
Coordinates: An ordered pair of numbers
used to locate a point on the Cartesian plane, where is the horizontal position (x-coordinate) and is the vertical position (y-coordinate). -
Distance Formula: A formula used to find the distance between two points
and on the Cartesian plane: -
Midpoint Formula: A formula used to find the midpoint of a line segment connecting two points
and : -
Slope: A measure of the steepness or incline of a line, defined as the ratio of the vertical change to the horizontal change between two points on the line:
-
Slope-Intercept Form: The equation of a line written as
, where is the slope and is the y-intercept (the point where the line crosses the y-axis). -
Point-Slope Form: The equation of a line that passes through a point
with slope : -
Y-Intercept: The point where a line crosses the y-axis, represented as
. -
X-Intercept: The point where a line crosses the x-axis, represented as
. -
Equation of a Circle: The equation of a circle with center
and radius is: - Collinear Points: Points that lie on the same straight line.
- Parallel Lines: Lines in the same plane that never intersect, having the same slope but different y-intercepts.
-
Perpendicular Lines: Lines that intersect at a right angle (90°). The slopes of perpendicular lines are negative reciprocals of each other:
- Isosceles Triangle: A triangle with two equal sides. In coordinate geometry, the distance formula can be used to check whether two sides of a triangle are equal.
- Equilateral Triangle: A triangle where all three sides are of equal length. Again, the distance formula is used to verify this.
-
Right-Angled Triangle: A triangle with one angle equal to 90°. The Pythagorean theorem is often used to verify right-angled triangles:
- Collinearity Test: A method of determining if three points are collinear by calculating the slopes of the line segments between them. If the slopes are equal, the points are collinear.
-
Distance between Parallel Lines: The shortest distance between two parallel lines is given by:
Where and are the constants in the equations of the lines, and and are coefficients of and in the general line equation. -
Centroid: The point where the medians of a triangle intersect, calculated as the average of the x-coordinates and y-coordinates of the triangle's vertices:
-
Pythagorean Theorem: A fundamental relation in a right-angled triangle between the lengths of the hypotenuse and the other two sides:
-
Inclination of a Line: The angle that a line makes with the positive x-axis, usually denoted by
. The slope of the line is related to the inclination angle by: -
General Form of a Line: The equation of a line in the form:
where , , and are constants. -
Vector Equation of a Line: The vector form of the equation of a line can be written as:
where is a point on the line, is the direction vector, and is a scalar. -
Area of a Triangle (using coordinates): The area of a triangle with vertices
, , and can be calculated using the formula: - Centroid of a Triangle: The point where the three medians of a triangle intersect. The centroid divides each median into a 2:1 ratio, with the longer part closer to the vertex.
- Circumcenter: The point where the perpendicular bisectors of the sides of a triangle intersect. It is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
- Orthocenter: The point where the altitudes of a triangle intersect. The altitude is the perpendicular segment from a vertex to the opposite side.
Straight Lines
Finding the Equation of a Straight Line
The equation of a line can be written in various forms. The slope-intercept form is:
Where:
is the slope. is the y-intercept.
The point-slope form is:
is the slope. is a point on the line.
Midpoint formula
Midpoint Formula:
Distance Formula:
Solved Problem
The point
Find the value of
Solution:
We use the midpoint formula:
Given that the midpoint is
Step 1: Equating the x-coordinates
Equate the x-coordinates:
Multiplying both sides by 2:
Solve for
Step 2: Equating the y-coordinates
Equate the y-coordinates:
Multiplying both sides by 2:
Solve for
Final Answer:
The values are:
Example :
Parallelogram Question
Three of the vertices of a parallelogram,
a) Find the midpoint of
b) Find the coordinates of
Solution:

Step 1: Find the midpoint of
The formula for the midpoint between two points
Substitute
Step 2: Find the coordinates of
Let the coordinates of
The midpoint of
Equating with the midpoint of
Step 3: Solve for and
For the x-coordinates:
Multiply both sides by 2:
For the y-coordinates:
Multiply both sides by 2:
Solve for
Final Answer:
The midpoint of
The coordinates of
Example :
Distance Formula Problem
The distance between two points
Find the two possible values of
Solution:
Step 1: Apply the distance formula
The distance formula is:
Substituting the coordinates
Simplify:
Step 2: Square both sides
Square both sides to remove the square root:
Step 3: Expand both terms
Expand both squares:
Step 4: Combine like terms
Combine the terms:
Step 5: Rearrange the equation
Rearrange the terms to form a quadratic equation:
Divide by 2:
Step 6: Solve the quadratic equation
Use the quadratic formula:
Substitute
Step 7: Calculate the two possible values of
The two values are:
Final Answer:
The two possible values of
Text Book Solutions
Step-by-Step Solutions for 5 Geometry Questions
Question 1a:
Calculate the lengths of the sides of the triangle
Points:
Step 1: Calculate
Using the distance formula:
Step 2: Calculate
Step 3: Calculate
Step 4: Verify if the triangle is right-angled
Check Pythagoras:
The triangle is right-angled.
Question 1b:
Calculate the lengths of the sides of the triangle
Points:
Step 1: Calculate
Step 2: Calculate
Step 3: Calculate
Step 4: Conclusion
Since
Question 2:
Show that triangle
Points:
Step 1: Calculate side lengths
Step 2: Verify right-angle
Pythagoras holds:
Step 3: Calculate the area
The area of a right-angled isosceles triangle is:
Question 3:
The distance between two points,
Step 1: Use the distance formula
Square both sides:
Step 2: Simplify the equation
Rearrange:
Divide by 2:
Step 3: Solve using the quadratic formula
Two possible values of
Question 4:
The distance between two points,
Step 1: Use the distance formula
Square both sides:
Step 2: Expand the terms
Combine terms:
Step 3: Solve the quadratic equation
Two possible values of
Question 5:
The point
Step 1: Use the midpoint formula
Midpoint formula:
Substitute
Step 2: Solve for
Equating x-coordinates:
Multiply both sides by 2:
Step 3: Solve for
Equating y-coordinates:
Multiply both sides by 2:
Final Answer:
Values of
Geometry Worksheet
Instructions:
Solve the following problems. Show all your work and justify your answers.
Question 1:
Calculate the lengths of the sides of triangle
Points:
Question 2:
The distance between two points
Question 3:
Show that the triangle with vertices
Question 4:
The point
Question 5:
The distance between two points
Question 6:
Find the length of the diagonal of a rectangle whose vertices are
Question 7:
Determine whether the points
Question 8:
The distance between two points
Question 9:
Find the area of triangle
Question 10:
The point
Answer Key
Answer 1:
Using the distance formula:
Since
Answer 2:
Using the distance formula:
Square both sides and simplify:
Solve the resulting quadratic to find
Answer 3:
Check side lengths:
Since
Area of triangle:
Answer 4:
Using the midpoint formula:
Solving, we get
Answer 5:
Using the distance formula:
Solve to find
Answer 6:
The diagonal of the rectangle is the distance between
Answer 7:
Find the slopes between
Since the slopes are equal, the points are collinear.
Answer 8:
Using the distance formula:
Solve to find
Answer 9:
Area of triangle:
Answer 10:
Using the midpoint formula:
Solving, we get
Question: 1

Step-by-Step Solutions
Question
i. Express
ii. State the smallest possible value of
iii. For the case where
Solution:
i. Express in the form
We begin by completing the square for the quadratic expression:
Step 1: Factor out the negative sign from the first two terms:
Step 2: Complete the square inside the parentheses. To complete the square, take half of the coefficient of
Step 3: Factor the perfect square trinomial:
Thus, the expression in the required form is:
Here,
ii. State the smallest possible value of for which is one-one.
For a quadratic function to be one-one, it must either be strictly increasing or strictly decreasing. From the completed square form
The smallest possible value of
iii. For the case where , find an expression for and state the domain of .
Given that
We will find the inverse function
Step 1: Subtract 4 from both sides:
Step 2: Multiply both sides by -1:
We get
Replace x with y, and y with x
Step 3: Take the square root of both sides:
Step 4: Solve for
The inverse function is:
The domain of the inverse function is restricted to values of
Final Answer:
i.
ii.
iii.
Question: 2

Step-by-Step Solutions
Question
The function
i. Express
ii. State the range of
iii. State the smallest possible value of
iv. For the value of
Solution:
i. Express in the form
We begin by completing the square for the quadratic expression:
Step 1: Factor out any constants from the first two terms:
Step 2: Complete the square inside the parentheses. To complete the square, take half of the coefficient of
Thus, the expression in the required form is:
Here,
ii. State the range of in terms of
From the expression
Thus, the range of
iii. State the smallest possible value of for which is one-one.
For a quadratic function to be one-one, we need to restrict the domain such that the function is either strictly increasing or strictly decreasing. From the expression
Therefore, the smallest possible value of
iv. For the value of , find an expression for and state the domain of , giving your answer in terms of .
We are given that
We will find the inverse function
Step 1: Subtract
Step 2: Take the square root of both sides (since the domain of
Step 3: Solve for
The inverse function is:
The domain of
Final Answer:
i.
ii.
iii.
iv.
Question: 3

Step-by-Step Solutions
Question:
The function
i. State the range of
ii. Copy the diagram and sketch the graph of
iii. Obtain expressions to define the function
Solution:
i. State the range of
First, we evaluate the range of each piece of the function:
- ForThus, the range of the entire function
ii. Copy the diagram and sketch the graph of
The graph of the inverse function
The graph of
iii. Obtain expressions to define the function , giving the set of values for which each expression is valid
We now need to find the inverse of each piece of
Final Answer:
i. Range of
ii. See graph of
iii. The inverse function is:


Step-by-Step Solutions
Question:
The function
i. Express in the form , where , , and are constants.
Solution:
We will complete the square to express the quadratic in the required form:
Now, complete the square for
Thus:
Simplifying:
Thus, the expression in the required form is:
ii. The function is defined for , where is a constant. It is given that is a one-to-one function. State the smallest possible value of .
In order for the quadratic function to be one-to-one, we need to restrict the domain so that the function is either strictly increasing or strictly decreasing. The function
To make
iii. Find the range of .
The range of
Thus, the minimum value of
Since the function is strictly increasing for
iv. Find the expression for and state the domain of .
We start by setting
Now, take the square root of both sides:
Since
Question

i. Express
Solution:
We will complete the square for the expression
Step 1: Take the quadratic part
Step 2: Substitute this back into the original expression:
Thus, the expression in the required form is:
ii. State the smallest possible value of .
Since
Thus, the smallest possible value of
iii. For the case where and , find and .
We are given that
So,
iv. Find an expression for and state the domain of .
To find the inverse function
Take the square root of both sides:
So:
Since
The domain of
Question

Step-by-Step Solution
i. Express in the form
We are given the function:
To express
Step 1: Factor out 2 from the quadratic terms:
Step 2: Complete the square for the expression inside the parentheses:
Substitute this back into the equation:
Step 3: Simplify the expression:
Thus, the expression in the required form is:
ii. State the range of
The function
The minimum value of
Therefore, the range of
iii. Find the set of values of for which
We are given that
Step 1: Set the inequality:
Step 2: Add 11 to both sides:
Step 3: Divide both sides by 2:
Step 4: Take the square root of both sides:
Step 5: Solve for
Therefore, the set of values of
iv. Find the value of the constant for which the equation has two equal roots
We are given that
First, express
Substitute
Simplify:
For the equation to have two equal roots, the expression inside the parentheses
Substitute
Thus,
Notes on Parallel and Perpendicular Lines

1. Parallel Lines
Definition: Parallel lines are two or more lines in the same plane that never intersect. They remain equidistant from each other at all points.
Properties of Parallel Lines:
- Parallel lines have the same slope.
- If two lines are parallel, their slopes are equal.
- The y-intercepts of parallel lines may be different.
Slope Condition for Parallel Lines:
Let the slopes of two lines be
Equation of Parallel Lines:
Two parallel lines can be represented as:
- First line:
- Second line:
If
Example:
Consider the lines
Perpendicular Lines

2. Perpendicular Lines
Definition: Perpendicular lines are two lines that intersect at a right angle (90°). Their slopes have a specific relationship: the product of their slopes is
Properties of Perpendicular Lines:
- Perpendicular lines intersect at 90°.
- The slopes of two perpendicular lines are negative reciprocals of each other.
- If one line has a slope of
, the slope of the perpendicular line is .
Slope Condition for Perpendicular Lines:
Let the slopes of two lines be
Equation of Perpendicular Lines:
If one line has the equation
Example:
Consider the lines
3. Parallel and Perpendicular Line Relationships
The relationship between parallel and perpendicular lines is important in both geometry and algebra:
- Parallel lines have equal slopes, but different y-intercepts (unless they are the same line).
- Perpendicular lines have slopes that are negative reciprocals of each other.
Summary of Conditions:
For two lines:
- If
, the lines are parallel. - If
, the lines are perpendicular.
4. Distance Between Two Parallel Lines
The shortest distance between two parallel lines
Example:
Consider the lines
Example: Find the distance of the point
Solution: The given line is
Comparing with the general equation of a line
Given point is
The distance of the given point from the given line is
Example: Estimate the distance between the two parallel lines
Solution: The distance between two parallel lines is given by
Here, the equations of parallel lines are
Slopes are the same
So, the distance between two parallel lines is given by,
Example:
Example 5: Let
- A)
- B)
- C)
- D)
Solution:
Slope of
Equation of line passing through
Hence, Option D is the answer.
5. Perpendicular Bisector
The perpendicular bisector of a line segment is the line that divides the segment into two equal parts and is perpendicular to the segment.
Equation of the Perpendicular Bisector:
To find the equation of the perpendicular bisector of the segment joining two points
- Find the midpoint of the segment:
- Find the slope of the line segment:
- The slope of the perpendicular bisector is the negative reciprocal of this slope:
Example:
Find the equation of the perpendicular bisector of the segment joining the points
Solution:
- Midpoint:
- Slope of the segment:
- Slope of the perpendicular bisector:
- Equation of the perpendicular bisector:
, or .
Question:
The coordinates of three points are
Answer:
If
Thus,
Example
The vertices of triangle
Answer:
Since angle
Cross-multiply to equate with
Clear the denominator:
Expand and simplify:
Upon re-evaluation (solving a quadratic equation derived from above expansions):
Solve for
Thus,

Reflections in Graphs: Detailed Notes
Understanding Reflections
Reflection: A transformation that "flips" a graph over a specified line, such as the x-axis or y-axis, producing a mirror image of the original graph. The most common types of reflections are:
- Reflection over the x-axis: This type of reflection is achieved by multiplying the entire function by
. If the original function is , then the reflected function is . - Reflection over the y-axis: This type of reflection is achieved by replacing
with in the function. If the original function is , then the reflected function is .
Example: Reflection Over the x-axis
Consider the function

The original function is:
The reflected function is:
When these two functions are graphed, they are mirror images of each other across the x-axis. This means that for every point
Key Observations:
- For every
value, the y-coordinates of the reflected graph are the negative of the y-coordinates of the original graph. - The shapes of the original graph and the reflected graph are identical, but they are positioned on opposite sides of the x-axis.

Example: Reflection Over the y-axis
Consider the function
The original function is:
The reflected function is:
This operation reflects the graph over the y-axis. The result is that for every point
Key Observations:
- The y-coordinates remain unchanged, but the x-coordinates are the negative of those in the original graph.
- The shapes of the original graph and the reflected graph are identical, but they are positioned on opposite sides of the y-axis.
Conclusion:
Reflections are a fundamental type of transformation that can be used to understand how graphs behave under certain operations. By reflecting a graph over the x-axis or y-axis, we can visualize how the graph would look if flipped across these axes. This concept is widely used in various mathematical applications and helps in understanding symmetry in functions.
Solved Examples
Reflections: Step-by-Step Solutions
Problem 2a:
Find the equation of the graph
Steps:
1. Original Function:
2. Reflection in the x-axis:
To reflect the graph over the x-axis, multiply the entire function by
Final Answer:
Problem 2b:
Find the equation of the graph
Steps:
1. Original Function:
2. Reflection in the y-axis:
To reflect the graph over the y-axis, replace
3. Simplify:
Since
Final Answer:
(Note: The reflection over the y-axis does not change the graph because
Problem 2c:
Find the equation of the graph
Steps:
1. Original Function:
2. Reflection in the y-axis:
To reflect the graph over the y-axis, replace
3. Simplify:
Simplify each term:
Final Answer:
Problem 2d:
Find the equation of the graph
Steps:
1. Original Function:
2. Reflection in the x-axis:
To reflect the graph over the x-axis, multiply the entire function by
3. Distribute
Distribute the negative sign across the function:
Final Answer:
Transformation Descriptions: Step-by-Step Solutions
Problem 3a:
Describe the transformation that maps the graph
Steps:
1. Compare the Original and Transformed Functions:
2. Identify the Transformation:
Notice that the entire function has been multiplied by
Final Answer:
The transformation that maps the original graph onto the new graph is a reflection over the x-axis.
Problem 3b:
Describe the transformation that maps the graph
Steps:
1. Compare the Original and Transformed Functions:
2. Identify the Transformation:
Notice that the sign of the linear term
Final Answer:
The transformation that maps the original graph onto the new graph is a reflection over the y-axis.
Problem 3c:
Describe the transformation that maps the graph
Steps:
1. Compare the Original and Transformed Functions:
2. Identify the Transformation:
Notice that the terms in the function have been reordered, with their signs reversed. This transformation involves both a reflection over the y-axis and a rearrangement of the terms.
Final Answer:
The transformation that maps the original graph onto the new graph is a reflection over the y-axis followed by rearranging the terms.
Problem 3d:
Describe the transformation that maps the graph
Steps:
1. Compare the Original and Transformed Functions:
2. Identify the Transformation:
Notice that the entire function has been multiplied by
Final Answer:
The transformation that maps the original graph onto the new graph is a reflection over the x-axis.
Question:
Find the equation of the perpendicular bisector of the line segment joining
Answer:
Step 1: Calculate the gradient of
Step 2: Find the gradient of the perpendicular bisector (negative reciprocal):
Step 3: Find the midpoint of
Step 4: Use the point-slope form to find the equation of the perpendicular bisector:
Substitute
Step 5: Simplify and expand:
Step 6: Multiply both sides by 2 to remove the fraction:
Final equation of the perpendicular bisector:
Question 1:
Find the equation of the line with:
- Gradient 2 passing through the point
- Gradient
passing through the point - Gradient
passing through the point
Solution:
The equation of the line is given by the point-slope form:
Substitute
Expand:
Simplify:
Solution:
Substitute
Expand:
Simplify:
Solution:
Substitute
Expand:
Simplify:
Question 2:
Find the equation of the line passing through each pair of points:
- Points
and - Points
and - Points
and
Solution:
Step 1: Calculate the gradient:
Step 2: Use point-slope form with
Expand:
Solution:
Step 1: Calculate the gradient:
Step 2: Use point-slope form with
Expand:
Simplify:
Solution:
Step 1: Calculate the gradient:
Step 2: Use point-slope form with
Expand:
Simplify:
Question 3:
Find the equation of the line:
- Parallel to the line
, passing through the point - Parallel to the line
, passing through the point - Perpendicular to the line
, passing through the point - Perpendicular to the line
, passing through the point
Solution:
The gradient of the required line is the same as that of
Expand:
Simplify:
Solution:
Step 1: Rearrange
Step 2: Use the same gradient
Expand:
Simplify:
Solution:
The gradient of the perpendicular line is the negative reciprocal of 2, which is
Expand:
Simplify:
Solution:
Step 1: Rearrange
Step 2: The gradient of the perpendicular line is
Expand:
Simplify:
Question 4:
Find the equation of the perpendicular bisector of the line segment joining the points:
- Points
and
Solution:
Step 1: Find the midpoint:
Step 2: Find the gradient of the line segment:
Step 3: The gradient of the perpendicular bisector is 2. Use point-slope form with the midpoint
Expand:
Simplify:
Question 5:
The line
Solution:
Step 1: Find the gradient of
Step 2: The gradient of
Step 3: Use the point-slope form to find the equation of
Expand:
Simplify:
Step 4: The gradient of
Expand:
Simplify:
Step 5: Solve the system of equations to find the coordinates of
Multiply through by 12 to eliminate fractions:
Solve for
Step 6: Substitute
Thus, the coordinates of
Question 6:
Given
- Find the equation of the line
. - Find the coordinates of
. - Find the area of triangle
.
Solution:
Step 1: Find the gradient of
Step 2: The gradient of the perpendicular line is
Expand:
Simplify:
Solution:
The point
Thus, the coordinates of
Solution:
The area of a triangle is given by:
Substitute
Thus, the area of triangle
Question 7:
The line
- Find the coordinates of
. - Find the equation of the line through
that is perpendicular to .
Solution:
Step 1: Substitute
Expand:
Step 2: Substitute
Thus, the coordinates of
Solution:
The gradient of
Expand:
Simplify:
Question 8:
The perpendicular bisector of the line joining
- Find the equation of the line
. - Find the coordinates of
and . - Find the length of
.
Solution:
Step 1: Find the midpoint of
Step 2: Find the gradient of
Step 3: The gradient of the perpendicular bisector is the negative reciprocal of
Expand:
Simplify:
Solution:
For
Thus,
For
Thus,
Solution:
Use the distance formula between
Thus, the length of

The standard form of the equation of a circle is:
Where:
is the center of the circle. is the radius of the circle.
Steps for Finding the Center and Radius
1. Recognize the Equation of a Circle:
- If the equation is of the form
, it represents a circle centered at the origin with radius . - If the equation is of the form
, the center is and the radius is .
Example A
Consider the equation
This is a standard form of a circle centered at the origin,
Thus, the center is
Example B
Now, consider the equation
From the equation, the center

Example C
Consider the equation
The center
Example D
Consider the equation
The center
Example E
Consider the equation
The center
Example F
Consider the equation
The center
Completed Table
The following table summarizes the centers and radii of the circles from the examples:
Conclusion
By examining the structure of the circle's equation, students can easily identify its center
Equation of a circle passing through two points

A is the point
Find the equation of the circle that has
Answer
The centre of the circle,
Radius of the circle,
Equation of the circle is
Therefore, the equation of the circle is:
General Equation of a circle
Find the centre and the radius of the circle
Answer
We answer this question by first completing the square.
Complete the square for both
Collect the constant terms together:
Now, compare with the equation
The center is
Important points to Remember

The Angle in a Semicircle is a Right Angle
Note: The angle formed at the circumference by a diameter of the circle is always a right angle (90 degrees).
Example Problem 1
Problem: In a circle with center
Solution:
Since
Example Problem 2
Problem: In a circle with center
Solution:
Since
The Perpendicular from the Center of a Circle to a Chord Bisects the Chord
Note: The perpendicular dropped from the center of a circle to any chord will always bisect the chord.
Example Problem 1
Problem: In a circle with center
Solution:
Since the perpendicular from the center of the circle bisects the chord,
Example Problem 2
Problem: In a circle with center
Solution:
The perpendicular from the center bisects the chord
The Tangent to a Circle at a Point is Perpendicular to the Radius at that Point
Note: The tangent line to a circle at any point on the circle is perpendicular to the radius drawn to the point of tangency.
Example Problem 1
Problem: In a circle with center
Solution:
By the property of tangents, the angle between the radius and the tangent at the point of tangency is 90 degrees. Therefore,
Example Problem 2
Problem: A circle with center
Solution:
The radius
1. Find the center and the radius of each of the following circles:
(a)
Solution:
(b)
Solution:
(c)
Solution:
(d)
Solution:
(e)
Solution:
(f)
Solution:
(g)
Solution:
(h)
Solution:
2. Find the equation of each of the following circles:
(a) Center
Solution:
(b) Center
Solution:
(c) Center
Solution:
(d) Center
Solution:
3. Find the equation of the circle with center (2, 5) passing through the point (6, 8).
Solution:
4. A diameter of a circle has its endpoints at A(-6, 8) and B(2, -4). Find the equation of the circle.
Solution:
Question
Find the equation of the line with gradient -4, and passing thru P(5m, 3m) intersects the x-Axis at A, and y-axis at B(i) Find the equation of the line
(ii) Find the area of the triangle AOB
(iii) Find the general form of the equation of a line
(iv) The line thru P perpendicular to AB intersects the x-axis at C, Check if the midpoint of PC lies on the line y = x
Where:
is the gradient of the line. is a point on the line.
Given:
- Gradient
- Point
Substitute these values into the equation of the line:
Simplifying:
Thus, the equation of the line is:
(ii) The coordinates of point
The coordinates of point
Finding the Area of Triangle
The area of triangle
Using the coordinates of
Simplifying this, we find:
Thus, the area of triangle
(iii) The equation of the line perpendicular to
(iv) The coordinates of point
(v) The midpoint of
Since
Question
Find the equation of the line with gradient -2, and passing thru P(3t, 2t) and intersects the x-Axis at A, and y-axis at B(i) Find the area of the triangle AOB in terms of t
The line thru P, perpendicular to AB, intersects the x-axis at C
(ii) Show that the midpoint of PC lies on y = x
Solution
1.
Solution: The equation of the line with gradient
Simplifying this equation:
The line intersects the x-axis at
Therefore, the area of triangle
2.
Solution: The equation of the line through
Simplifying:
This line intersects the x-axis at
Since the midpoint is
Question

Problem:
The diagram shows the graph of
(a) State the range of
Solution:
Step 1: Analyze the function
We are given the function:
The cosine function
Step 2: Apply the transformations
- Scaling by
gives the range: - Adding
shifts the range:
Step 3: Simplify the range
The final range becomes:
Thus, the range of

Solution for Question (b):
The value of
This means there is no vertical scaling applied to the function
The transformation is an identity transformation, meaning the curve remains unchanged under this mapping.
Solution for Question (c):
The equation of the curve reflected in the x-axis is given by:
Simplifying this gives:
Thus, the equation in the form
Problem:
The diagram shows the graph of
(a) State the range of
Solution:
Step 1: Analyze the function
We are given the function:
The cosine function
Step 2: Apply the transformations
- Scaling by
gives the range: - Adding
shifts the range:
Step 3: Simplify the range
The final range becomes:
Thus, the range of
Question

Solution for the Equation of a Circle:
A circle passes through the points
Find the equation of the circle.
Step 1: Find the Midpoint of and the Perpendicular Bisector
The midpoint of
The gradient of
The gradient of the perpendicular bisector of
The equation of the perpendicular bisector of
Step 2: Find the Midpoint of and the Perpendicular Bisector
The midpoint of
The gradient of
The gradient of the perpendicular bisector of
The equation of the perpendicular bisector of
Step 3: Solve the Two Equations to Find the Centre of the Circle
Solving equations (1) and (2) gives:
The centre of the circle is
Step 4: Find the Radius of the Circle
The radius is the distance from the centre
Step 5: Equation of the Circle
The equation of the circle is:
Question:
Find the equation of the circle with center(2,5) passing thru the point (6,8)Solution
The general equation of a circle with centerHere, the center is
We are also given that the circle passes through the point
Therefore, the equation of the circle is:
Hence, the equation of the circle is
Question

Solution
We are given:
and the conditions:
Part (a): Solving for
We start with
Substitute into
Since
Next, use
Substitute into
Since
Part (b): Find
Using
Simplifying:
Therefore,
Exploring Unique and Thought-Provoking Knowledge
This section explores questions from various fields that provide deep insights and an intellectual edge.
Worked Examples
Q1
What was the primary reason for the collapse of the Bronze Age civilizations around 1200 BCE?
Solution:
- The collapse is attributed to a combination of factors, including natural disasters, invasions by the Sea Peoples, internal social upheavals, and a breakdown in trade networks.
- This disrupted the supply of tin and copper necessary for making bronze, leading to the downfall of major civilizations such as the Mycenaeans, Hittites, and Egyptians.
Q2
What is the "Ship of Theseus" paradox, and what does it question about identity?
Solution:
- The "Ship of Theseus" paradox asks whether a ship that has had all its components replaced remains fundamentally the same ship.
- This paradox challenges the concept of identity, questioning whether an object that undergoes gradual change can still be considered the same object.
Q3
What is Gödel's Incompleteness Theorem, and how does it impact mathematical systems?
Solution:
- Gödel's Incompleteness Theorem states that in any consistent mathematical system, there are statements that are true but cannot be proven within that system.
- This means that no mathematical system can be both complete and consistent, implying that some truths will always lie beyond formal proof.
Q4
What is the "Fermi Paradox," and what does it question about extraterrestrial life?
Solution:
- The Fermi Paradox questions why, despite the high probability of extraterrestrial life, we have not yet encountered any evidence of other civilizations.
- This paradox highlights the apparent contradiction between the vastness of the universe and the lack of observational evidence of alien life.
Q5
What is the Sapir-Whorf Hypothesis, and how does it influence our understanding of language and thought?
Solution:
- The Sapir-Whorf Hypothesis posits that the structure of a language influences its speakers' cognition and worldview.
- It suggests that people who speak different languages perceive and think about the world differently because language shapes thought processes.
Q6
What is the "Red Queen Hypothesis," and how does it relate to evolutionary biology?
Solution:
- The Red Queen Hypothesis suggests that organisms must constantly adapt and evolve to survive in an ever-changing environment where other species are also evolving.
- This concept emphasizes continuous evolutionary arms races between species.
Q7
What is the significance of Göbekli Tepe, and how does it challenge traditional views of human civilization?
Solution:
- Göbekli Tepe is significant because it predates the advent of agriculture and permanent settlements.
- This challenges the traditional view that complex societies developed only after the establishment of agriculture and suggests that organized religion and monumental architecture may have preceded farming.
Q8
What is the "Dunning-Kruger Effect," and how does it influence people's perception of their abilities?
Solution:
- The Dunning-Kruger Effect is a cognitive bias in which people with low ability or knowledge overestimate their competence, while those with high competence tend to underestimate their abilities.
- This effect highlights the lack of self-awareness in individuals regarding their own skills.
Q9
What is the "Paradox of Thrift," and how does it relate to macroeconomic theory?
Solution:
- The Paradox of Thrift suggests that while saving money is beneficial for individuals, if everyone saves more during a recession, overall demand will fall, leading to lower total savings and a weaker economy.
- This paradox illustrates the conflict between individual rationality and collective outcomes in economics.
Q10
What is the significance of the "Arnolfini Portrait" by Jan van Eyck, and what are its key symbolic elements?
Solution:
- The "Arnolfini Portrait" is significant for its intricate detail and use of symbolism.
- Key elements include the mirror reflecting the scene, the dog symbolizing loyalty, and the use of light to create depth and realism, often interpreted as a visual contract or depiction of marriage.
Algebra Week 5 Teaching Plan
Worked Examples
Q1 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Complete the square:
- The vertex is
. - The formula for the axis of symmetry is
. For this equation, and , so: - The axis of symmetry is
.
Q2 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Recognize that this is a perfect square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q3 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Factor out the negative sign and complete the square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q4 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Factor out the -2 and complete the square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q5 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Complete the square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q6 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Complete the square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q7 (Find the Vertex and Axis of Symmetry for )
Given:
Solution:
- Factor out the -4 and complete the square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q8: Find the Vertex and Axis of Symmetry for
Given:
Solution:
- Factor out the -9 and complete the square:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q9: Find the Vertex and Axis of Symmetry for
Given:
Solution:
- This is already a perfect square form:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Q10: Find the Vertex and Axis of Symmetry for
Given:
Solution:
- This is already a perfect square form:
- The vertex is
. - Using the formula for the axis of symmetry,
, with and : - The axis of symmetry is
.
Quadratic Equations-finding a and b
Worked Examples
Q1 (Write in the form , where , , and are integers)
Given:
Solution:
- First, factor out the coefficient of
from the first two terms: - Now complete the square by adding and subtracting the square of half the coefficient of
. Half of 2 is 1, and : - Simplify the expression:
- Distribute the 4:
- Combine the constants:
- Thus, the quadratic function in the form
is: where , , and .
Intersection of a Quadratic Curve and a Line
Q2 (Find any points of intersection between the curve and the line )
Given:
- Curve
has the equation . - Line
has the equation .
Solution:
- To find the points of intersection, set the equations for the curve and the line equal to each other:
- Rearrange the equation to form a standard quadratic equation:
- Factor the quadratic equation:
- Solving for
, we get: - Substitute
back into the equation of the line to find the corresponding value of : - Therefore, the point of intersection is
.
Quadratic Equations-Conditions for No Real Roots
Worked Examples
Q1: Show that for the equation having no real roots
Given:
Solution:
- The general condition for a quadratic equation to have no real roots is that the discriminant must be less than zero. The discriminant
of a quadratic equation is given by: - For the given equation
, the coefficients are: - Substituting these values into the discriminant formula:
- For the equation to have no real roots, the discriminant must be less than zero:
- Simplifying the inequality:
Dividing both sides by 4: - Thus, we have shown that
.
Q2: Find the minimum value of the function , giving your answer in terms of
Given:
Solution:
- To find the minimum value of the function, we complete the square.
- First, factor the quadratic and linear terms:
- Complete the square by adding and subtracting the square of half the coefficient of
. Half of 4 is 2, and : - The minimum value of
occurs when . Therefore, the minimum value of the function is: - Thus, the minimum value of the function is
, and it occurs when .
Q3: Sketch the graph of , labelling any points where the graph crosses the coordinate axes and the turning point
Given:
Solution:
- To sketch the graph, we need to find:
- The x-intercepts (roots of the quadratic)
- The y-intercept (value of
when ) - The turning point (vertex of the parabola)
- To find the x-intercepts, set
and solve the quadratic equation: - We can use the quadratic formula:
where , , and . - Substituting the values:
- This gives two solutions:
- Therefore, the x-intercepts are
and . - To find the y-intercept, set
: - The y-intercept is
. - We can find the turning point using the formula for the vertex
. For the given equation, and : - Substitute
into the equation to find the corresponding -value: Simplifying this will give the value of at the turning point.
Step 1: Find the x-intercepts
Step 2: Find the y-intercept
Step 3: Find the turning point
