Mathematics Notes

Form 2 Mathematics – PYTHAGORAS THEOREM

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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 x c)

(a+b) (a+b) = ab + c2

[a2 + 2ab + b2] = ab + c2

a2 + ab + b2 = ab + c2

a2 + b2 = c2

a2 + b2 = c2

Pythagoras theorem

Examples

1.The side s of a triangle containing the right angle have length of 5cm and 12cm.

Find the length of the hypotenuse

Solution

C2= a2 +b2

C2 = 52 + 122

C2 = 25+ 144

C2 = 169

C =

C= 13cm

∴The length of the hypotenuse = 13cm.

  1. In figure below if AC =17cm, BC = 8cm, and CD = 12cm 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 = 8cm

∴ r2 = 82+ 82
r2 = 64 +64

∴r =11.31cm

  1. Given triangle ABC, where B = 900.Find the lengths of the sides which are not given

 

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

Solution:

X2 = 152 +82

X2 = 225 +64

X =

X = 17km

∴ He is 17km from his starting point

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

Solution:

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