MATHEMATICS O LEVEL(FORM TWO) NOTES – PYTHAGORAS THEOREM

PYTHAGORAS THEOREM

PYTHAGORAS THEOREM
Pythagoras theorem is used to solve problems involving right-angled triangles.

Statement:

In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

Shown below

Required to prove: c² = a² + b²

Construction:

Joining L and N. Considering the trapezium PQLN:

Area of the trapezium

But area of trapezium = area Δ PKN + area Δ KQL + area Δ KLN

(a+b)(a+b) = ab + (c × c)

(a+b)(a+b) = ab + c²

[a² + 2ab + b²] = ab + c²

a² + ab + b² = ab + c²

a² + b² = c²

Examples

1. The sides of a triangle containing the right angle have lengths of 5 cm and 12 cm.
   Find the length of the hypotenuse.

Solution

C² = a² + b²

C² = 5² + 12²

C² = 25 + 144

C² = 169

C =

C = 13 cm

∴ The length of the hypotenuse = 13 cm.

2. In the figure below, if AC = 17 cm, BC = 8 cm, and CD = 12 cm, find AD.

Solution:

EXERCISE

1. Calculate the unknown side of the following triangle.

SOLUTION:

17² = 15² + b²

b² = 17² – 15²

b² = 289 – 225

b =

b = 8 cm

∴ r² = 8² + 8²

r² = 64 + 64

∴ r = 11.31 cm

2. Given triangle ABC, where B = 90°, find the lengths of the sides which are not given.

Find the area of the triangle and the length of the perpendicular from C to B.

3. A man travels 15 km due north and then 8 km due west. How far is he from his starting point?

Solution:

X² = 15² + 8²

X² = 225 + 64

X =

X = 17 km

∴ He is 17 km from his starting point.