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Grade 4 · Foundation and Higher · free GCSE maths worksheet

Pythagoras' theorem: finding the hypotenuse

Squaring, adding and square rooting.

14 original questions, 42 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Triangle ABC is a right-angled triangle with the right angle at B.

AB = 6 cm and BC = 8 cm.

Work out the length of AC.

ABC6 cm8 cmDiagram NOT accurately drawn
2 marks
Show answer
10 cm
2

A right-angled triangle has shorter sides of length 5 cm and 12 cm.

Work out the length of the hypotenuse.

2 marks
Show answer
13 cm
3

Triangle PQR is a right-angled triangle with the right angle at Q.

PQ = 7 cm and QR = 9 cm.

Work out the length of PR.

Give your answer correct to 1 decimal place.

PQR7 cm9 cmDiagram NOT accurately drawn
2 marks
Show answer
11.4 cm
4

In triangle PQR, angle Q = 90°, PQ = 7 cm and QR = 9 cm.

Work out the length of PR. Give your answer correct to 3 significant figures.

3 marks
Show answer
11.4 cm
5

A ship sails 24 km due north and then 10 km due east.

Work out the straight-line distance of the ship from its starting point.

3 marks
Show answer
26 km
6

Here are the side lengths of two triangles.

  1. Triangle A: 9 cm, 12 cm, 15 cm
  2. Triangle B: 7 cm, 8 cm, 12 cm

Is each triangle right-angled? Show your working.

4 marks
Show answer
a) Yes, 9² + 12² = 15² b) No, 7² + 8² = 113 but 12² = 144
7

A rectangular field is 60 m long and 45 m wide. Kyle walks from one corner to the opposite corner. He can walk along two sides of the field, or straight across the diagonal.

How much shorter is the diagonal route than walking along the two sides?

3 marks
Show answer
30 m
8

A is the point with coordinates (2, 3).

B is the point with coordinates (10, 9).

Work out the length of AB.

2 marks
Show answer
10
9

A triangle has vertices A(2, 1), B(10, 1) and C(2, 7). M is the midpoint of AB and N is the midpoint of AC.

  1. Write down the coordinates of M and of N.
  2. Work out the length of MN.
4 marks
Show answer
a) M(6, 1), N(2, 4) b) 5
10

A rectangle is 9 cm long and 6 cm wide.

Work out the length of a diagonal of the rectangle.

Give your answer correct to 1 decimal place.

2 marks
Show answer
10.8 cm
11

A right-angled triangle has sides 6 cm, 8 cm and 10 cm (the hypotenuse). A semicircle is drawn outwards on each side, using that side as its diameter.

Show that the area of the semicircle on the hypotenuse is equal to the sum of the areas of the other two semicircles.

4 marks
Show answer
Semicircle on 10 cm: 12.5π; on 6 cm: 4.5π; on 8 cm: 8π; 4.5π + 8π = 12.5π
12

A triangle has sides of length 10 cm, 24 cm and 26 cm.

Show that the triangle is a right-angled triangle.

3 marks
Show answer
102 + 242 = 100 + 576 = 676 = 262, so by the converse of Pythagoras' theorem the triangle is right-angled
13

A square has sides of length 5 cm.

A circle is drawn through all four vertices of the square.

Calculate the circumference of the circle. Give your answer correct to 3 significant figures.

5 cmDiagram NOT accurately drawn
4 marks
Show answer
22.2 cm
14

A rectangle measures 8 cm by 15 cm.

A circle is drawn that passes through all four vertices of the rectangle, so a diagonal of the rectangle is a diameter of the circle.

Work out the circumference of the circle.

Give your answer correct to 1 decimal place.

4 marks
Show answer
53.4 cm

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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