Pythagoras' Theorem
Senior 2 · Mathematics
Pythagoras' Theorem
Lesson 1 of 4
22 min
A builder in Rubavu wants the corners of a new classroom to be perfect right angles. A phone shop advertises a "15 cm screen" (measured along the diagonal). An engineer needs the length of a cable from the top of a mast to the ground. All three problems are solved by one of the most famous results in mathematics: Pythagoras' theorem, which links the three sides of every right-angled triangle.
A right-angled triangle has one angle of 90°. The side opposite the right angle is the hypotenuse; it is always the longest side. The other two sides, which form the right angle, are called the legs. In triangle ABC with the right angle at C, the hypotenuse is AB.
Draw a right-angled triangle with legs 3 and 4 and a hypotenuse of 5, and build a square on each side.
The areas are 9, 16 and 25, and 9 + 16 = 25. The two squares on the legs together have exactly the same area as the square on the hypotenuse. This is true for every right-angled triangle.
In a right-angled triangle with legs a and b and hypotenuse c:
a2 + b2 = c2
The legs are 6 cm and 8 cm. Then c2 = 62 + 82 = 36 + 64 = 100, so c = √100 = 10 cm.
The hypotenuse is 13 cm and one leg is 5 cm. Then 52 + b2 = 132, so b2 = 169 − 25 = 144 and b = 12 cm.
To find a leg you subtract; to find the hypotenuse you add.
The legs are 5 m and 7 m. c2 = 25 + 49 = 74, so c = √74 ≈ 8.60 m. Leave it as √74 m if an exact answer is asked for (Unit 1).
Tip
Exam technique. First find the hypotenuse: it is opposite the right-angle mark. Put it alone on one side, c2 = a2 + b2. After you calculate, check that the hypotenuse is the longest side. If your "leg" comes out longer than the hypotenuse, you added when you should have subtracted.
Find the hypotenuse of a right-angled triangle with legs 9 cm and 12 cm.
Solution: c2 = 81 + 144 = 225, so c = 15 cm.
A right-angled triangle has hypotenuse 10 cm and one leg 6 cm. Find the other leg.
Solution: b2 = 100 − 36 = 64, so b = 8 cm.
Find the diagonal of a rectangle 5 cm by 12 cm.
Solution: The diagonal is the hypotenuse of a right-angled triangle with legs 5 and 12: √(25 + 144) = √169 = 13 cm.
Find the height of an equilateral triangle with side 6 cm, exactly and to 1 decimal place.
Solution: The height splits the base into 3 + 3. h2 = 36 − 9 = 27, so h = √27 = 3√3 ≈ 5.2 cm.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
