Chapter 8: Similar Triangles

Pythagoras Theorem. (B audhayan Theorem)

Theorem 8.3: - In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Given: - is a right angle of

To Prove:-

Construction:- is drawn

Proof:- In

Or, AB² = AC X AD ---------------------(i)

Similarly

Or, BC² = AC X CD ---------------------(ii)

Adding (i) and (ii) we get

AB² + BC² = AC X AD + AC X CD

= AC X (AD + CD)

= AC X AC

= AC²

Or, AC² = AB² + BC²

Theorem 8.4 (Converse of Pythagoras Theorem): - In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Given:- In

To prove:-

Construction:- A triangle PQR is constructed such that PQ = AB, QR = BC and

Proof:- In

Or, PR² = AB² +BC²--------------(i) [PQ = AB, QR = BC]

But AC² = AB² + BC² ------------(ii) (given)

Or, PR = AC

Or,

Hence

Example 13. Determine whether the triangle having sides (2a – 1) cm, and (2a + 1) cm is a right angled triangle.

Sol:- Let AB = (2a - 1) cm,

AC = (2a + 1) cm

is a right angled triangle.