Form 2 Mathematics – 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                           

 
 

                                     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.

2.  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

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

 
 

                         

                

 
                    

              

                           
 

   
 

3.   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: