Unit 3 Section 2 : Calculating the Hypotenuse

If we know the two shorter lengths in a right-angled triangle, we can use Pythagoras' Theorem to find the hypotenuse.

Example Question

Look at the triangle below.

We want to find the length of the hypotenuse (h).

Pythagoras' Theorem tells us:
      The square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
In this case, this means that h² is equal to 5² + 12².

So the length of the hypotenuse (h) can be worked out as follows:
h² = 5² + 12² (work out the squares of the two shorter sides)
h² = 25 + 144 (add the squares of the two shorter sides together)
h² = 169 (square root both sides to find the value of h)
h = 13
We can now see that the length of the hypotenuse (h) is 13m.

Practice Questions

(a) Look at the triangle below.

We want to find the length of the hypotenuse (d).

Pythagoras' Theorem tells us:
      The square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
In this case, this means that d² is equal to (1.5)² + (2)².

Work out the length of the hypotenuse (d) using the method above, then click Click on this button below to see the correct answer to see if you are correct.

(b) Look at the triangle below.

We want to find the length of the hypotenuse (s).

Work out the length of the hypotenuse (s) to 1 decimal place, then check your answer below.

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
Calculate the length of the hypotenuse in each of the triangles below.
(a) cm
(b) cm
(c) mm
(d) cm
Question 2
Calculate the length of the hypotenuse in each of the triangles below.
Give your answers correct to 1 decimal place.
(a) cm
(b) cm
(c) cm
(d) m
Question 3
A rectangle has sides of lengths 5cm and 10cm.

How long is the diagonal of the rectangle?
Give your answer correct to 1 decimal place.
cm

Question 4
A square has sides of length 6cm.

How long is the diagonal of the square?
Give your answer correct to 1 decimal place.
cm

Question 5
The diagram below shows a wooden frame that is to be part of the roof of a house.
(a) Use Pythagoras' Theorem in triangle PQR to find the length PQ.
m
(b) Calculate the length QS.
m
(c) Calculate the total length of wood needed to make the frame.
m
Question 6
The diagram below shows an isosceles triangle with a base of length 4cm and perpendicular height 8cm.
(a) Calculate the length, x, of one of the equal sides.
cm
(b) Calculate the perimeter of the triangle.
cm
Question 7
The diagram below shows a vertical flagpole of height 5.2m, with a rope tied to the top.
When the rope is pulled tight, the bottom end is 3.8m from the base of the flagpole.
Calculate the length of the rope.
m
Question 8
A rectangular lawn is 12.5m long and 8m wide.
Matthew walks diagonally across the lawn from one corner to the other.
He returns to the first corner by walking round the edge of the lawn.

How much further does he walk on his return journey?
m

Question 9
Look at the two rectangles below.

Which rectangle has the longer diagonal?

You have now completed Unit 3 Section 2
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Produced by A.J.Reynolds February 2003
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