Applications of Pythagoras' theorem
Apply Pythagoras' theorem to solve real-world problems including finding the diagonal of a rectangle and the distance between two points.
Worked examples
Find the diagonal of a rectangle
Straightforward
Problem
A rectangle has length cm and width cm. Find the length of the diagonal.
1
Identify the right-angled triangle. The diagonal of a rectangle is the hypotenuse of a right-angled triangle formed by the length and width.
Legs: cm and cm. Hypotenuse: .
2
Write Pythagoras' theorem.
3
Substitute the known values.
4
Take the square root to find the diagonal.
Answer
10 cm
Find the distance between two points
Moderate
Problem
Find the distance between the points and .
1
Find the horizontal distance by subtracting the -coordinates.
2
Find the vertical distance by subtracting the -coordinates.
3
These two distances are the legs of a right-angled triangle. Apply Pythagoras' theorem.
4
Take the square root to find the distance.
Answer
10 units
Real-world problem: ladder against a wall
Challenging
Problem
A ladder m long leans against a vertical wall. The foot of the ladder is m from the base of the wall. How high up the wall does the ladder reach?
1
Identify the sides of the right-angled triangle. The ladder is the hypotenuse; the wall height and ground distance are the shorter sides.
Hypotenuse m, base m, height .
2
Write Pythagoras' theorem with the unknown height.
3
Evaluate the squares and rearrange.
4
Take the square root to find the height.
Answer
12 m
Practise
Q1·Straightforward
A rectangle has length cm and width cm. Find the length of the diagonal.
Explanation
Q2·Straightforward
A rectangle has length cm and width cm. Find the length of the diagonal.
Explanation
Q3·Straightforward
A rectangle has length cm and width cm. Find the length of the diagonal.
Explanation
Q4·Straightforward
A rectangle has length cm and width cm. Find the length of the diagonal.
Explanation
Q5·Moderate
Find the distance between the points and .
Explanation
Horizontal distance , vertical distance .
Q6·Moderate
Find the distance between the points and .
Explanation
Horizontal distance , vertical distance .
Q7·Moderate
Find the distance between the points and .
Explanation
Horizontal distance , vertical distance .
Q8·Moderate
Find the distance between the points and .
Explanation
Horizontal distance , vertical distance .
Q9·Challenging
A ladder m long leans against a vertical wall. The foot of the ladder is m from the base of the wall. How high up the wall does the ladder reach? Give your answer in metres.
Explanation
The ladder, wall, and ground form a right-angled triangle. The ladder ( m) is the hypotenuse.
Q10·Challenging
A rectangular television screen is cm wide and cm tall. Find the length of the diagonal, correct to 2 decimal places.
Explanation
Q11·Challenging
A ship travels km north and then km east. How far is the ship from its starting point? Give your answer in kilometres.
Explanation
The ship's path forms a right-angled triangle with legs km and km.
Q12·Challenging
A ramp rises m vertically over a horizontal distance of m. Find the length of the ramp surface in metres.
Explanation