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8.3 Using Pythagoras’ theorem

8.3 Using Pythagoras’ theorem

$e longest side of a right-angled triangle is called the hypotenuse. Hypotenuse
$e hypotenuse is the side that is opposite the right angle.
Now look at this triangle. $e length of the hypotenuse is labelled a and
the lengths of the other two sides are b and c.
a
Pythagoras’ theorem states that in any right-angled triangle, the square of b
the hypotenuse is equal to the sum of the squares of the other two sides.
2
For this triangle: a = b + c 2 c
2
You can use this formula to solve problems involving right-angled triangles.

Worked example 8.3
a A right-angled triangle has a base length of 1.2 m and a perpendicular height of 0.9 m.
What is the length of the hypotenuse of the triangle?
b A ladder is 5 m long. Dave rests the ladder against a vertical brick wall. The foot of the ladder is 1.5 m
horizontally from the base of the wall. How far up the wall does the ladder reach?

a Start by drawing a triangle to represent the problem. Write the dimensions
0.9 m that you know on the triangle.
1.2 m

a = Label the sides of the triangle a, b and c. Label the hypotenuse a and the
b = 0.9 other two sides b and c. It doesn’t matter which is which.

c = 1.2
a = b + c Write down the formula, then substitute in the numbers that you know.
2
2
2
a = 0.9 + 1.2 Solve the equation to work out the value of a. Take it one step at a time.
2
2
2
a = 0.81 + 1.44
2
a = 2.25
2
a = 225. Use your calculator to work out the square root.
a = 1.5 m Remember to write the correct units (metres) with your answer.
b Start by drawing a triangle to represent the problem. Write the dimensions
a = 5 m that you know on the triangle.
b =
Label the sides of the triangle a, b and c.
c = 1.5 m
a = b + c Write down the formula, then substitute in the numbers that you know.
2
2
2
5 = b + 1.5 Solve the equation to work out the value of b. Take it one step at a time.
2
2
2
25 = b + 2.25
2
b = 25 − 2.25
2
b = 22.75
2
b = 22 75. Use your calculator to work out the square root.
b = 4.77 m (2 d.p.) If your answer isn’t exact, round it to two decimal places. Put the units in
your answer.
8 Constructions and Pythagoras’ theorem 81