Finding unknown sides
Use Pythagoras' theorem to find a shorter side of a right-angled triangle given the hypotenuse and one other side.
Worked examples
Find a shorter side (integer result)
Straightforward
Problem
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
1
Write Pythagoras' theorem.
2
Rearrange to find the unknown side .
3
Substitute the known values.
4
Take the square root to find .
Answer
12 cm
Find a shorter side (decimal result)
Moderate
Problem
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg, rounded to 2 decimal places.
1
Write the rearranged form of Pythagoras' theorem.
2
Substitute the known values.
3
Take the square root and round to 2 decimal places.
Answer
9.80 cm
Multi-step: find a side then calculate perimeter
Challenging
Problem
A rectangle has a diagonal of cm and one side of cm. Find the perimeter of the rectangle.
1
Recognise that the diagonal is the hypotenuse of a right-angled triangle with legs equal to the rectangle's sides.
2
Substitute the known values to find the missing side.
3
Take the square root.
4
Calculate the perimeter.
Answer
42 cm
Practise
Q1·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
Q2·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
Q3·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
Q4·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
Q5·Moderate
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg, rounded to 2 decimal places.
Explanation
Q6·Moderate
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg, rounded to 2 decimal places.
Explanation
Q7·Moderate
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg, rounded to 2 decimal places.
Explanation
Q8·Moderate
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg, rounded to 2 decimal places.
Explanation
Q9·Challenging
A rectangle has a diagonal of cm and a length of cm. Find the width of the rectangle.
Explanation
Using Pythagoras' theorem:
Q10·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? Round your answer to 2 decimal places.
Explanation
Q11·Challenging
A right-angled triangle has a hypotenuse of cm and its two shorter sides are in the ratio . Find the length of the shorter side.
Explanation
Let the sides be and .
Shorter side cm.
Shorter side cm.
Q12·Challenging
A right-angled isosceles triangle has a hypotenuse of cm. Find the length of each equal side, rounded to 2 decimal places.
Explanation
Let each equal side be .