Pythagoras' theorem GCSE lesson: hypotenuse, shorter side, the converse and distance

Pythagoras: practice

Show your working on paper. Find the hypotenuse first in every question. Round only at the end, as each question says. The stretch is optional.

Diagnostic question: choose and explain

1. A right-angled triangle has hypotenuse 15 cm. One of its other sides is 9 cm. How long is the third side?
   1. 17.5 cm
   2. 144 cm
   3. 12 cm
   4. 6 cm

Practice questions

1. The two sides that meet at the right angle are 8 cm and 15 cm. Find the hypotenuse, in cm.
2. A right-angled triangle has hypotenuse 25 cm and one other side 7 cm. Find the third side, in cm.
3. The two sides that meet at the right angle are 3 cm and 8 cm. Find the hypotenuse in cm, to 1 decimal place.
4. Without a calculator: the two sides that meet at the right angle are both 5 cm. Mark the length of the hypotenuse, in cm, on the number line.They drag a marker on a number line from 0 to 12.
5. Each line gives the three sides of a triangle. Sort them: is the triangle right-angled? Test the longest side against the other two.They sort 8 items into Right-angled and Not right-angled.
6. Triangle PQR has its right angle at Q. Which side is the hypotenuse?
   1. PQ
   2. PR
   3. QR
7. Find the distance between the points (2, 3) and (8, 11).
8. A ladder 5 m long leans against a wall. Its foot is 1.4 m from the wall, on flat ground. How high up the wall does it reach, in m?
9. An isosceles triangle has sides 10 cm, 10 cm and 12 cm. Find its area, in cm².
10. Stretch: the points (1, k) and (4, 7) are 5 apart. So (k − 7)² + 3² = 5². Solve it line by line, and give both values of k.They solve (k − 7)² + 3² = 5² line by line, each line checked as they write it.

Support: For question 1: the two sides you are given meet at the right angle, so you want the hypotenuse. Square both, add, then take the square root. For a shorter side, square both, subtract the smaller from the larger, then take the root. Check: the hypotenuse is always the longest side.

Pythagoras: exit

Show your working for both.

1. A right-angled triangle has hypotenuse 20 cm and one other side 12 cm. Find the third side, in cm.
2. A triangle has sides 6 cm, 9 cm and 11 cm. Is it right-angled? Show how you know.

Exit check

A shorter side found, and one triangle tested, show who subtracts and who checks for a right angle.

Find a shorter side from the hypotenuse and one other side.

Pythagoras: lengths, tests and distances

For each triangle, say first which side is the hypotenuse, and whether you add or subtract. Round only at the end, as the question says. For question 5, sketch the two points on a grid with the triangle between them.