Pythagoras' Theorem
Label the Triangle
Draw a line to match each term to its description.
The Formula
Circle the correct statement about Pythagoras' theorem.
Pythagoras' theorem:
To find hypotenuse c:
To find shorter side a:
Find the Hypotenuse
Use c² = a² + b².
a = 3, b = 4
a = 5, b = 12
a = 8, b = 15
a = 6, b = 8
Find a Shorter Side
Use a² = c² − b².
c = 10, b = 6
c = 13, b = 5
c = 25, b = 20
Pythagorean Triples
Draw a line to match each pair to the correct hypotenuse.
Right-Angled or Not?
Check a² + b² = c². Sort each triangle.
Pythagoras in Real Life
Draw a diagram and show all working.
A 5 m ladder leans against a wall. The base is 2 m from the wall. How high up the wall does the ladder reach? Round to 2 decimal places.
A rectangular field is 48 m long and 36 m wide. What is the length of the diagonal path across it?
3D Pythagoras — Diagonal of a Box
The space diagonal of a rectangular prism = √(l² + w² + h²). Show all working.
A box is 3 cm long, 4 cm wide and 12 cm tall. Calculate the length of the longest diagonal (corner to corner through the box).
A room is 5 m long, 3 m wide and 2.4 m high. What is the length of the diagonal from one bottom corner to the opposite top corner? Round to 2 decimal places.
Pythagorean Triples — Extended
A Pythagorean triple is a set of three whole numbers satisfying a² + b² = c². Draw a line to match each pair of legs to the correct hypotenuse.
Distance Between Two Coordinate Points
Use d = √((x₂ − x₁)² + (y₂ − y₁)²) to find the distance.
Points (0, 0) and (3, 4)
Points (1, 1) and (4, 5)
Points (0, 0) and (5, 12)
Navigation and Bearing Problems
Draw a diagram for each problem. Show all working.
A ship sails 12 km due East and then 5 km due North. How far is the ship from its starting point? In what direction (roughly) is it from the start?
Two towns are shown on a grid. Town A is at (2, 3) and Town B is at (8, 11). What is the straight-line distance between them?
Which Formula Applies?
Sort each problem: do you need to find the hypotenuse or a shorter side?
Find the Hypotenuse — Set A
Use c = √(a² + b²). Show all working.
a = 9, b = 12. Find c.
a = 20, b = 21. Find c.
a = 7, b = 24. Find c.
Find a Shorter Side — Set A
Use a = √(c² − b²). Show all working.
c = 17, b = 8. Find a.
c = 26, b = 10. Find a.
c = 29, b = 20. Find a.
Checking Right Triangles
Check whether each set of side lengths forms a right-angled triangle. Show working.
Sides: 9, 40, 41. Right-angled?
Sides: 11, 12, 16. Right-angled?
Sides: 5, 8, √89. Right-angled?
Real-Life Problem — Ladders
Draw a diagram, label the right angle and hypotenuse, then solve.
A 10 m ladder leans against a wall with its base 3 m from the wall. How high up the wall does it reach? Round to 2 d.p.
A ladder reaches 8 m up a wall. The ladder is 10 m long. How far is its base from the wall?
Coordinate Geometry — Distance Formula
Use d = √((x₂ − x₁)² + (y₂ − y₁)²). Show working.
Find the distance from (2, 1) to (6, 4).
Find the distance from (−3, 0) to (1, 3).
Find the distance from (0, −2) to (5, 10).
Distance Formula Steps
Put the steps in order for finding the distance between (1, 2) and (4, 6).
Coordinate Perimeter
Plot the points, draw the shape, and calculate its perimeter.
Points: A(0,0), B(4,0), C(4,3). Calculate the perimeter of triangle ABC.
Find the Hypotenuse — Non-Integer Answers
Give the exact surd form, then a decimal rounded to 2 d.p.
a = 5, b = 7. Find c.
a = 4, b = 9. Find c.
a = 6, b = 11. Find c.
Find a Shorter Side — Non-Integer Answers
Give the exact surd form, then round to 2 d.p.
c = 12, b = 7. Find a.
c = 15, b = 9. Find a.
c = 20, b = 13. Find a.
Area Using Pythagoras
Find the unknown height using Pythagoras, then calculate the area.
An isosceles triangle has equal sides of 10 cm and a base of 12 cm. Find the height, then find the area.
An equilateral triangle has side length 8 cm. Find the height using Pythagoras, then find the area. Round to 2 d.p.
3D Pythagoras — Extended
Apply d = √(l² + w² + h²). Show all working.
Find the space diagonal of a rectangular box 6 m × 8 m × 10 m.
A room is 4 m long, 3 m wide and 2.5 m high. A mouse runs from one bottom corner to the diagonally opposite top corner. How far?
Multi-Step Navigation Problem
Draw a diagram for each problem. Show all working.
A hiker walks 8 km North, then 6 km East. How far from the start?
A boat sails 15 km West and then 8 km South. How far from the start?
Match the Problem Type
Match each description to the correct formula.
Converse of Pythagoras
The converse says: if a² + b² = c², the triangle is right-angled. Use this to check.
A builder uses measurements 60 cm, 80 cm and 100 cm to check a corner. Is the corner square?
Stakes are placed with distances 5 m, 7 m and 9 m. Is the shape rectangular? Explain.
Circles and Pythagoras
The perpendicular from the centre to a chord bisects it. Use Pythagoras to find the unknown.
A circle has radius 13 cm. A chord is 10 cm from the centre. Find the chord length.
A circle has radius 17 cm. The chord has length 16 cm. How far is the chord from the centre?
Ramps and Gradients
Use Pythagoras to solve each ramp problem. Show all working.
A wheelchair ramp rises 0.6 m over a horizontal distance of 8 m. What is the length of the ramp surface?
A ski slope is 120 m long and rises 30 m vertically. What is the horizontal distance covered?
Pythagorean Triples Investigation
Use the formula: for integers m > n > 0, the triple is (m² − n², 2mn, m² + n²).
Use m = 2, n = 1 to generate a triple. Verify it works.
Use m = 3, n = 2 to generate a triple. Verify it works.
Why must m and n be different positive integers?
Design Challenge: Roof Truss
A triangular roof truss has a horizontal base of 8 m and a vertical height of 3 m.
The roof has two equal sloping sides. Calculate the length of each rafter. Round to 2 d.p.
If each rafter needs a 10% safety margin, what total timber length is needed for both rafters?
Error Analysis
Find and fix the errors in this student's working.
Student: 'Find c when a = 6, b = 8. c = 6 + 8 = 14.' What is wrong? What is the correct answer?
Student: 'Find a when c = 10, b = 6. a = √(10² + 6²) = √136.' What is wrong? What is the correct answer?
Pythagoras and Surds
Simplify each surd answer fully.
Find c when a = 4, b = 6. Simplify fully.
Find c when a = 3, b = √7. Simplify fully.
Find a when c = √50, b = √18. Simplify fully.
Reflection: Pythagoras in the Real World
Write your response in the box.
Name three real-world situations where Pythagoras' theorem is used. For one of them, write a problem and solve it fully.
Pythagoras Around the Home
Apply Pythagoras' theorem to real measurements.
- 1Measure the length and width of a room. Calculate the diagonal distance from one corner to the opposite corner. Check by measuring directly with a tape measure.
- 2Measure the height and width of a door frame. Calculate the diagonal length. This tells you the longest object that can fit through the door lying flat.
- 3Research: what are the dimensions of a standard football (soccer) pitch (105 m × 68 m)? What is the length of the diagonal?