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8.3 Using Pythagoras’ theorem

) Exercise 8.3

1 Work out the length of the hypotenuse in each triangle.
The first two have been started for you.

a a = a = b + c 2 b a = b + c 2
2
2
2
2
a = a =
b = 6 m a = b = 6 m 2 2 2 9 mm 9 mm 9 mm 2 2 2
b = 6 m
9 mm
b = 6 m
a = 9 + 12
a = 6 + 8
c = 8 cm
12 mm
c = 8 cm a = 36 + 64 12 mm 12 mm
c = 8 cm
2
12 mm
c = 8 cm
c d
7 cm 5 m7 cm 7 cm 5 m 5 m
7 cm
5 m
10 cm 10 cm 10 cm 9 m 9 m 9 m
9 m
10 cm
2 Work out the lengths of the sides marked in each triangle.
The first two have been started for you.

a a = 5 cm a = b + c 2 b 2.6 m a = b + c 2
2
2
2
2
a = 5 cm
2.6 m
b = b = 2 a = 5 cm 1 m 2.6 m 1 m 2 2 2
2
2
a = 5 cm b = 5 = b + 4 1 m 2.6 = 1 + c
2.6 m
b = c = 4 cm 25 = b + 16 1 m
c = 4 cm
2
c = 4 cm
2
c = 4 cm b = 25 – 16
15 m
c 15 m 13 cm 25 cm d 13 cm 25 cm
15 m
25 cm
15 m 25 cm 13 cm
9 m 13 cm 9 m
9 m
9 m
3 A rectangle is 12 cm long and 5 cm wide. Draw diagrams to help you solve
Work out the length of a diagonal of the rectangle. these problems.
4 Isaac walks 8 km north and then 12 km east.
How far is Isaac from his starting point?
16 cm 12 cm
5 The diagram shows a triangle inside a circle with centre O.
The lengths of the shorter sides of the triangle are 12 cm and 16 cm. O
Work out the area of the circle.
Summary
You should now know that: You should be able to:

+ An inscribed shape fits inside a circle with all its + Use a straight edge and compasses to construct
vertices (corners) on the circumference of the the perpendicular from a point on a line and the
circle. perpendicular from a point to the line.
+ Pythagoras’ theorem can only be used to solve + Use a straight edge and compasses to inscribe
problems in right-angled triangles. The theorem squares, equilateral triangles and regular
states that: a = b + c , where a is the hypotenuse hexagons and octagons by constructing equal
2
2
2
and b and c are the two shorter sides. divisions of a circle.
+ Know and use Pythagoras’ theorem to solve
two-dimensional problems involving right-angled
triangles.
82 8 Constructions and Pythagoras’ theorem