Finding a shorter side
Rearrange Pythagoras' theorem to find an unknown shorter side (leg) of a right-angled triangle.
Worked examples
Find a shorter side using Pythagoras' theorem
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
Substitute the known values. The hypotenuse and one leg .
3
Evaluate the squares.
4
Rearrange to isolate .
5
Take the square root to find .
cm
Answer
cm
Find the height a ladder reaches
Moderate
Problem
A 10 m ladder leans against a wall. The foot of the ladder is 4 m from the base of the wall. Find the height the ladder reaches up the wall, correct to 2 decimal places.
1
Identify the sides. The ladder is the hypotenuse ( m), the horizontal distance is one leg ( m), and the height is the unknown leg.
2
Evaluate the squares.
3
Rearrange to isolate .
4
Take the square root and round to 2 decimal places.
m
Answer
m
Find the height of an isosceles triangle
Challenging
Problem
An isosceles triangle has two equal sides of cm and a base of cm. Find the perpendicular height of the triangle.
1
Draw a perpendicular from the apex to the base. It bisects the base into two equal halves.
Half-base cm
2
This creates a right-angled triangle with hypotenuse cm (the equal side) and one leg cm (the half-base). Write Pythagoras' theorem.
3
Evaluate the squares.
4
Rearrange to isolate .
5
Take the square root.
cm
Answer
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
Using : , so cm.
Q2·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
, so cm.
Q3·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
, so cm.
Q4·Straightforward
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
, so cm.
Q5·Moderate
A 10 m ladder leans against a wall. The foot of the ladder is 4 m from the base of the wall. Find the height the ladder reaches up the wall. Give your answer correct to 2 decimal places.
Explanation
, so m.
Q6·Moderate
A rectangular field is 30 m wide. A diagonal fence across the field is 34 m long. Find the length of the field.
Explanation
, so m.
Q7·Moderate
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the length of the other leg.
Explanation
, so cm.
Q8·Moderate
A ramp is 6 m long and rises 3 m vertically. Find the horizontal distance covered by the ramp. Give your answer correct to 2 decimal places.
Explanation
, so m.
Q9·Challenging
An isosceles triangle has two equal sides of cm and a base of cm. Find the perpendicular height of the triangle.
Explanation
The perpendicular bisects the base into two halves of cm. The right-angled triangle has hypotenuse cm and one leg cm. , so cm.
Q10·Challenging
A square has a diagonal of cm. Find the side length of the square. Give your answer correct to 2 decimal places.
Explanation
cm.
Q11·Challenging
A right-angled triangle has a hypotenuse of cm and one leg of cm. Find the area of the triangle in .
Explanation
Other leg: , so cm. Area cm.
Q12·Challenging
A rhombus has a side length of cm. One of its diagonals is cm long. Find the length of the other diagonal. (The diagonals of a rhombus bisect each other at right angles.)
Explanation
Half the known diagonal cm. The other half-diagonal satisfies , so cm. The full diagonal cm.