Distance, Gradient & Midpoint
Match the Formula
Draw a line from each concept to its correct formula.
Calculate the Gradient
Use m = (y2 - y1) / (x2 - x1) to find the gradient of each line segment.
A(1, 2) and B(3, 6):
P(0, 5) and Q(4, 1):
C(-2, 3) and D(4, 3):
E(2, -1) and F(2, 7):
Positive, Negative, Zero or Undefined?
Circle the correct type of gradient for each line description.
A line going up from left to right
A horizontal line
A vertical line
A line going down from left to right
Calculate the Midpoint
Find the midpoint of each segment using M = ((x1+x2)/2, (y1+y2)/2).
A(2, 4) and B(8, 10):
P(-3, 1) and Q(5, 7):
C(0, 0) and D(6, -4):
E(-5, -2) and F(3, 6):
Calculate the Distance
Use d = sqrt[(x2-x1)^2 + (y2-y1)^2] to find the exact distance.
A(0, 0) and B(3, 4):
P(1, 1) and Q(4, 5):
C(-1, 2) and D(5, 10):
Mixed Practice
For the segment joining A(-2, 1) and B(4, 9), find each value.
Gradient of AB:
Midpoint of AB:
Length of AB (exact surd form):
Application: Map Coordinates
A town map uses a grid. Answer the questions below.
Town Hall is at (2, 3) and the Library is at (10, 9). Find the straight-line distance (leave as a surd if needed).
A new bus stop will be placed at the midpoint between Town Hall and the Library. What are the coordinates of the bus stop?
Collinear Points Challenge
Three points are collinear if the gradient between any two pairs is equal.
Show whether A(1, 2), B(3, 6) and C(5, 10) are collinear by calculating gradients of AB and BC.
Gradient of Vertical and Horizontal Lines
Calculate the gradient of each segment and describe the type of line.
A(2, 5) and B(8, 5): gradient = ___, type of line =
P(3, 1) and Q(3, 9): gradient = ___, type of line =
Explain why the gradient of a vertical line is undefined.
Midpoint Formula Application
Use the midpoint formula to solve each problem.
The midpoint of AB is M(4, 7). Point A is at (2, 3). Find the coordinates of point B.
A bridge is built between two points on opposite banks of a river: (10, 20) and (30, 40). At what coordinates is the midpoint of the bridge?
Distance Formula Word Problems
Set up coordinates and use the distance formula to solve.
Two friends live at positions A(3, 4) and B(9, 12) on a map grid where 1 unit = 1 km. How far apart do they live?
Three cities are at P(0, 0), Q(6, 8) and R(6, 0). Find the perimeter of the triangle PQR.
Identifying Gradient Properties
Circle the correct statement about each pair of lines.
Line 1 has gradient 2. Line 2 has gradient 2.
A horizontal line has gradient:
A vertical line has gradient:
All Three Measures -- Mixed Practice
For each pair of points, calculate the gradient, midpoint and exact distance.
A(0, 0) and B(5, 12): gradient = ___, midpoint = ___, distance = ___
P(-3, 4) and Q(5, -2): gradient = ___, midpoint = ___, distance = ___
Gradient Calculations — Set A
Calculate the gradient for each pair of points.
A(2, 4) and B(6, 8): m =
A(0, 5) and B(4, 1): m =
A(-2, -3) and B(4, 9): m =
A(3, 7) and B(3, 2): m =
A(-5, 4) and B(3, 4): m =
Midpoint Calculations — Set A
Find the midpoint of each segment.
A(2, 4) and B(8, 10): M =
A(-3, 1) and B(5, 7): M =
A(0, 0) and B(10, -6): M =
A(-4, -2) and B(4, 8): M =
A(1.5, 3) and B(4.5, 9): M =
Find the Missing Endpoint
Given the midpoint and one endpoint, find the other endpoint.
Midpoint M(5, 6), one end A(2, 4). Find B.
Midpoint M(0, 0), one end A(3, -5). Find B.
Midpoint M(-1, 2), one end A(4, 8). Find B.
Midpoint M(3, 3.5), one end A(-1, 2). Find B.
Match Points to Gradient
Match each pair of points to its gradient.
Which Gradient is Steeper?
Circle the line with the steeper gradient.
Which is steeper?
Which falls more steeply?
Which is steepest?
Gradient from a Graph
Read the gradient from each described line.
A line passes through (0, 2) and (4, 6). Gradient =
A line passes through (1, 5) and (3, 1). Gradient =
A line passes through (0, -3) and (5, 7). Gradient =
A horizontal line through (2, 4). Gradient =
A vertical line through (-3, 1). Gradient =
Distance on a Number Line
Understand distance formula as an extension of 1D distance.
Points A(2) and B(8) on a number line: distance = |8 - 2| = ___
Points A(3, 0) and B(9, 0) — horizontal segment: distance = ___
Points A(0, 4) and B(0, 10) — vertical segment: distance = ___
Why is the distance formula (involving square root) needed for diagonal distances?
Verify a Right Angle Using Distance
Use the distance formula to verify whether a triangle has a right angle.
Triangle with vertices A(0,0), B(4,0), C(0,3). Find all three side lengths and verify it is right-angled.
Triangle with vertices P(1,1), Q(5,1), R(3,5). Is it right-angled? Find all side lengths and check.
Coordinate Geometry — Applying All Three Formulas
For each pair of points, find the distance, midpoint and gradient.
C(1, 2) and D(7, 10): distance = ___, midpoint = ___, gradient = ___
E(-4, 3) and F(2, -5): distance = ___, midpoint = ___, gradient = ___
G(0, -3) and H(9, 9): distance = ___, midpoint = ___, gradient = ___
J(-5, -1) and K(3, 5): distance = ___, midpoint = ___, gradient = ___
Gradient in Context
Calculate the gradient in each real-world scenario.
A ramp rises 1.2 m over a horizontal distance of 6 m. Gradient =
A ski slope drops 400 m vertically over 1200 m horizontally. Gradient =
A road rises 80 m for every 2 km along the road. Gradient (as a fraction) =
Is a gradient of 1/5 steeper than a gradient of 1/3? Explain how you know.
Collinear Points
Three points are collinear (on the same line) if the gradient between any two pairs is equal. Check whether each set of three points is collinear.
A(1, 2), B(3, 6), C(5, 10). Are they collinear? Show working.
P(0, 1), Q(2, 4), R(4, 6). Are they collinear? Show working.
X(-2, -3), Y(0, 1), Z(3, 7). Are they collinear? Show working.
Coordinate Geometry in Your Neighbourhood
Apply distance, midpoint and gradient to a local map.
- 1Open Google Maps and find two landmarks near your home. Estimate their coordinates on a grid (using street intersections). Calculate the distance and midpoint.
- 2Measure a slope near your home (a driveway, ramp, or hill). Estimate the rise and run. Calculate the gradient and compare it to Australian building standards.
- 3Draw a coordinate grid of your local area on 1 cm grid paper. Place your home at the origin. Mark 5 locations, calculate distances between them, and find midpoints.
Perpendicular Gradients
Find the gradient of the perpendicular line.
Line has gradient 2. Perpendicular gradient =
Line has gradient -1/3. Perpendicular gradient =
Line has gradient 4/5. Perpendicular gradient =
Verify: multiply your answer by 4/5. Result should be -1: ___
Line through A(0,0) and B(3,6) has gradient m = ___. A perpendicular line through B has gradient ___.
Perpendicular Bisector
Find the perpendicular bisector of each segment.
Segment AB: A(2, 1), B(8, 5). Find the midpoint M and perpendicular gradient. The bisector passes through M with gradient ___.
Segment CD: C(0, 6), D(8, 2). Find midpoint and perpendicular gradient.
Sort Lines by Gradient Category
Place each line in the correct column.
Coordinate Geometry — Triangle Properties
Use distance, gradient and midpoint to investigate a triangle.
Triangle: A(0,0), B(6,0), C(3,4). (a) Find all three side lengths. (b) Is the triangle isosceles? (c) Find the midpoint of each side.
Show that the triangle in the previous question is isosceles (two sides equal) by comparing lengths.
Gradient in Linear Equations
Connect gradient to the equation of a line y = mx + b.
Line: y = 3x - 2. State the gradient and y-intercept.
Line through (0, 4) with gradient 2. Write the equation y = mx + b.
Line through A(1, 3) and B(4, 9). Find the gradient, then the equation y = mx + b.
What is the equation of the horizontal line through (5, 7)?
Distance, Gradient, Midpoint — Comprehensive Practice
This set covers all three formulas in applied contexts.
A diagonal of a rectangle has endpoints (1, 2) and (9, 8). Find the length and midpoint of this diagonal.
A ladder leans from the ground at (0, 0) to a wall at point (3, 7). What is the length of the ladder? What is the gradient?
Points A(2, k) and B(5, 1) have a distance of 5. Find k.
A segment has gradient 2/3 and midpoint (5, 4). Find two possible pairs of endpoints.
Coordinate Geometry — Exam Practice
Work independently for 20 minutes. Show all working.
Find the distance between P(3, -1) and Q(-5, 5).
Find the midpoint of segment PQ (same points as above).
Find the gradient of PQ.
Midpoint M(4, 5), one endpoint A(1, 3). Find the other endpoint B.
A triangle has vertices O(0,0), A(8,0), B(4,6). Find: all side lengths, whether it is isosceles, and the midpoint of OB.
Explain in words what the midpoint formula calculates and why it works.
Distance and Midpoint — Final Practice Set
Solve these in preparation for the next topic.
Find the perimeter of a quadrilateral with vertices A(0,0), B(5,0), C(5,4), D(0,4).
Is point M(3, 5) the midpoint of A(1, 2) and B(5, 8)? Verify algebraically.
Find all points on the x-axis that are 10 units from A(0, 6).
Two friends start at (0, 0) and walk to (6, 8) and (8, -6) respectively. Who walks further?
Coordinate Geometry in Technology
Explore how coordinate geometry appears in technology.
- 1Open a spreadsheet program. In column A enter x-coordinates 0 to 10. In column B enter y = 2x + 3. Plot a line chart. What is the gradient and y-intercept?
- 2On graph paper, plot a route you take regularly (e.g. home to school). Assign coordinates to key intersections. Calculate the total distance using the distance formula.
- 3Research 'gradient descent in machine learning'. Write a 3-sentence explanation of how gradient is used to train neural networks.
Coordinate Geometry — Mastery Check
Demonstrate mastery by completing all questions without assistance.
State the distance, midpoint and gradient formulas from memory.
Given A(-3, 2) and B(5, -4): find distance, midpoint, gradient.
A segment has midpoint (4, 1) and one endpoint (7, 5). Find the other endpoint.
Explain the connection between the distance formula and Pythagoras' theorem.
Sketch a line with gradient -2 and y-intercept 3. Mark two points on it with exact coordinates.
Distance Formula — Applied Problems
Solve these using the distance formula.
Point P(3, y) is exactly 5 units from A(0, 1). Find all possible values of y.
Point Q(x, 4) is exactly 10 units from the origin. Find all possible values of x.
Show that the points A(0,0), B(4,3), C(8,0) form an isosceles triangle.
Coordinate Geometry — Quadrilateral Investigation
Use distance and gradient to classify the quadrilateral with vertices A(0,0), B(4,0), C(6,3), D(2,3).
Calculate the length of each side: AB, BC, CD, DA.
Calculate the gradient of each side: AB, BC, CD, DA.
Are any sides parallel (equal gradients)? Are any sides equal in length?
What type of quadrilateral is ABCD? (e.g. parallelogram, trapezium, rectangle)
Line Equation from Two Points
Use the gradient formula to write the equation of the line through each pair of points.
Through A(1, 3) and B(3, 7): m = ___, equation:
Through P(0, 5) and Q(2, 1): m = ___, equation:
Through X(-2, -1) and Y(4, 5): m = ___, equation:
Create Your Own Coordinate Geometry Problem
Design a problem using the distance, midpoint or gradient formula.
Write a word problem that requires the distance formula. Include the solution.
Write a problem that requires finding a missing endpoint given a midpoint. Include the solution.
Write a problem involving gradient in a real-world context. Include the solution.
Coordinate Geometry — Final Comprehensive Review
Complete all questions to demonstrate full mastery.
A(2, -1) and B(8, 7): find distance, midpoint, gradient.
Midpoint of AB is M(5, 4). One endpoint is A(2, 1). Find B.
Show that A(0,0), B(5,0), C(5,12), D(0,12) form a rectangle using gradient and distance.
A is at (1, 2) and B is at (7, 10). Find the equation of the perpendicular bisector of AB.
Three vertices of a parallelogram are P(1,1), Q(5,1), R(7,4). Find the fourth vertex S.
Gradient Investigations
Answer these investigative questions about gradient.
A road rises 50 m over 1 km of horizontal distance. Express the gradient as a fraction and as a percentage.
A river flows from (0, 200) to (100, 150) on a topographic grid. What is the gradient of the river's bed?
A line has gradient 3/4. Starting from point (4, 2), what are the coordinates of the next three points on the line (increasing x by 4 each time)?
Coordinate Geometry Proof
Use coordinate geometry to prove geometric results.
Prove that A(0,0), B(4,2), C(2,4) form an isosceles right triangle. (Find all three side lengths and check two are equal and that d_AB^2 + d_BC^2 = d_AC^2 or similar.)
Prove that the diagonals of the rectangle A(0,0), B(6,0), C(6,4), D(0,4) bisect each other. (Show that both diagonals have the same midpoint.)
Distance and Midpoint — Exam-Style Questions
Work for 20 minutes on these exam-style questions.
Find the exact length of the segment joining A(-3, 5) and B(4, -1).
A segment PQ has midpoint (3, -2) and P = (7, 4). Find Q.
Show that A(0,3), B(4,5), C(6,1) form an isosceles triangle with AB = CB.
A point T lies on the y-axis and is equidistant from A(3, 1) and B(-3, 5). Find T.
Gradient Between Multiple Points
For each set of three collinear points, verify they are collinear and find the common gradient.
A(0, 1), B(2, 4), C(4, 7). Verify collinear and state gradient.
P(-3, 0), Q(0, 3), R(3, 6). Verify collinear and state gradient.
X(1, 5), Y(3, 3), Z(5, 1). Verify collinear and state gradient.
Map Coordinate Geometry
Apply the three formulas to a real map.
- 1On Google Maps, find your local shopping centre, school, and library. Assign each a coordinate (e.g. use street block numbers as x and y). Calculate the distances between each pair.
- 2Find two landmarks in your city and calculate the gradient of the straight line between them on a map. Is the path uphill or downhill?
- 3Look up 'Great Circle Distance' — it is the formula for distance on a sphere (like Earth). How does it relate to the flat coordinate geometry distance formula?
Distance, Gradient, Midpoint — Summary and Connections
Write a summary connecting all three formulas.
Explain in 2-3 sentences how the distance formula is derived from Pythagoras' theorem.
Explain how the midpoint formula gives the 'average' position of two points.
Explain what gradient measures and give two real-world examples.
Describe a situation where you would need to use all three formulas together.
Midpoint Applications
Solve midpoint problems in context.
A bridge spans from pier A at (10, 0) to pier B at (50, 0). A support cable is attached at the midpoint. What are the coordinates of the attachment point?
Two friends live at coordinates (2, 7) and (8, 3) on a grid map. They agree to meet at the point exactly halfway between their houses. What are the coordinates of the meeting point?
The midpoint of segment PQ is (5, 4). If P = (2, 1), find the coordinates of Q.
Gradient Calculation Steps
Arrange the steps for calculating gradient between two points in the correct order.
Identify the Correct Formula
Circle the correct formula for each situation.
Distance between (x₁, y₁) and (x₂, y₂)
Midpoint of (x₁, y₁) and (x₂, y₂)
Gradient between (x₁, y₁) and (x₂, y₂)
Match: Gradient to Line Type
Match each gradient description to the correct line type.
Coordinate Geometry Proof
Use coordinates to prove geometric properties.
The vertices of a triangle are A(0,0), B(4,0), and C(2,4). Calculate the length of all three sides. Is it isosceles? Justify your answer.
Find the midpoints of AB, BC, and CA from the triangle above. What shape do the midpoints form? Support your answer with gradient calculations.
Sort by Gradient Steepness
Sort these gradients from least steep to most steep (ignore sign).
Coordinate Geometry Extended Problem
Solve this multi-step coordinate geometry problem.
Points A(1, 2), B(5, 4), C(3, 8), and D(−1, 6) form a quadrilateral. Calculate the lengths of all four sides.
Find the midpoints of the two diagonals AC and BD. What do you notice?
Based on your answers, what type of quadrilateral is ABCD? Explain.
Design a Coordinate Geometry Problem
Create your own coordinate geometry question.
Choose two points and calculate the distance, gradient, and midpoint. Show full working.
Now write a word problem that uses your two points in a real-world context (e.g. a map, a ramp, a building).
Swap with a partner (or parent) and solve each other's problems.
Gradient Interpretation
Circle the correct interpretation of each gradient.
A line has gradient m = 3
A line has gradient m = −1/2
A line has gradient m = 0
Gradient Walk
Investigate gradients in your local environment.
- 1Find a ramp, hill, or staircase near your home. Measure or estimate the rise and run to calculate the gradient. Is it steep or gentle?
- 2Using a printed local map with a grid, mark two landmarks and calculate the straight-line distance between them using the scale.
- 3Stand at one end of your street and look at a distant house. Estimate the height difference and horizontal distance, then calculate the gradient.
Explain Coordinate Geometry in Your Own Words
Write explanations for a younger student.
Imagine explaining the distance formula to a Year 7 student. Write a simple explanation using Pythagoras' theorem.
Explain what gradient means using a real-life analogy (like a hill, a ramp, or a staircase).
Describe what the midpoint formula finds and why it is useful.