Pythagoras Theorem class 10th

Pythagoras theorem:-

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. 

c2 = b2 + a2

b=Perpendicular of right triangle

a= Base of right triangle c= hypotenuse of right triangle For Example:-

Solution:-

Perpendicular of right triangle =b=5

Base of right triangle = a Hypotenuse of right triangle=c= 13

According to formula

 a2 = b2 - c2

 a = √ (b2 - c2)

a = √ (132 - 52)

a = √ (169 - 25)

a = √ (144)

a = 12 Ans.

Theorem 6.8
In a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two side.
Given:- A right triangle ABC right angled at A.

 Prove that:- BC2 = AB2 + AC2
Construction:- AD⏊BC (in figure)
Proof:-In △ABD and △ACB 
∠B =∠B (common)
and, ∠ADB =∠CAB (90°)
Now, △ABD〜△CBA 
             (AA similarity) 

so,  AB/BC=BD/AB 
(Corresponding parts of congruent triangle)

or, AB2= BC.BD........(1)

In △ACD and △ACB 
∠C =∠C (common)
and, ∠ADC =∠CAB  (90°)
Now, △ABD〜△CBA (AA similarity) 

so,  AC/BC=CD/AC 
(Corresponding parts of congruent triangle)

or, AC2= BC.CD........(2)

Adding (1) and (2), we have:

AB2+AC2=BC.BD+BC.CD
AB2+AC2=BC(BD+CD)

AB2+AC2=BC.BC

AB2+AC2=BC2

BC2=AB2+BC2



BC2=AB2+AC2




Hence, theorem proved