Finding the hypotenuse
Use Pythagoras' theorem to find the length of the hypotenuse in right-angled triangles.
Worked examples
Finding the hypotenuse from two legs
Straightforward
Problem
A right-angled triangle has legs of length cm and cm. Find the length of the hypotenuse.
1
Write Pythagoras' theorem.
2
Substitute the known values.
3
Take the square root of both sides.
cm
Answer
cm
Length of a ladder leaning against a wall
Moderate
Problem
A ladder leans against a wall with its base m from the wall and reaching m up the wall. How long is the ladder?
1
Identify the right-angled triangle. The ladder is the hypotenuse, the horizontal distance is one leg ( m), and the vertical height is the other leg ( m).
m, m, find
2
Apply Pythagoras' theorem and substitute.
3
Take the square root.
m
Answer
The ladder is m long.
Diagonal of a square
Challenging
Problem
A square has a side length of m. Find the length of its diagonal, to 2 decimal places.
1
Recognise that the diagonal of a square divides it into two right-angled isosceles triangles. Each triangle has two legs equal to the side length of the square.
m, m, find
2
Apply Pythagoras' theorem.
3
Take the square root. The result is a surd — simplify and then evaluate to 2 decimal places.
m
Answer
m
Practise
Q1·Straightforward
A right-angled triangle has legs of length cm and cm. Find the length of the hypotenuse.
Explanation
Using Pythagoras' theorem: , so cm.
Q2·Straightforward
A right-angled triangle has legs of length cm and cm. Find the length of the hypotenuse.
Explanation
, so cm.
Q3·Straightforward
A right-angled triangle has legs of length m and m. Find the length of the hypotenuse.
Explanation
, so m.
Q4·Straightforward
A right-angled triangle has legs of length cm and cm. Find the length of the hypotenuse.
Explanation
, so cm.
Q5·Moderate
A ladder leans against a wall with its base m from the wall and reaching m up the wall. How long is the ladder? Give your answer in metres.
Explanation
, so m.
Q6·Moderate
A rectangular paddock is m wide and m long. Find the length of the diagonal fence line, in metres.
Explanation
, so m.
Q7·Moderate
A ramp rises m over a horizontal distance of m. Find the length of the ramp surface, to 2 decimal places.
Explanation
, so m.
Q8·Moderate
A path is laid diagonally across a rectangular park that is m wide and m long. Find the length of the path, to 2 decimal places.
Explanation
, so m.
Q9·Challenging
A square has a side length of cm. Find the length of its diagonal, to 2 decimal places.
Explanation
The diagonal is the hypotenuse with both legs equal to cm. , so cm.
Q10·Challenging
A rectangle is m long and m wide. Find the length of its diagonal, to 2 decimal places.
Explanation
, so m.
Q11·Challenging
A right-angled triangle has legs of length cm and cm. Find the length of the hypotenuse, to 2 decimal places.
Explanation
, so cm.
Q12·Challenging
A right trapezium has parallel sides of length cm (bottom) and cm (top). The perpendicular height is cm. Find the length of the slant side, in centimetres.
Explanation
The horizontal leg of the right triangle equals the difference in the parallel sides: cm. The vertical leg is cm. So , giving cm.