Chapter 8: Similar Triangles

Exercise - 15

1. The perpendicular AD on the base BC of a intersects BC at D so that DB = 3CD. Prove that 2AB² = 2AC² + BC².

2. P and Q are points on the side CA and CB respectively of a right angled at C.

Prove that AQ² + BP² = AB² + PQ^2.

3. In , if AD is the median, Show that AB² + AC² = 2(AD² + BD²)

4. PQR is an isosceles right triangle, right angled at R. Prove that PQ² = 2PR².

5. In a is an acute angle and Prove that AC² = AB² + BC² – 2BC. BD.

6. In the adjoining figure, find the length of BD, If

7. Prove that the altitude of an equilateral triangle of side

8. P and Q are the midpoint of the sides CA and CB respectively of right angled at C. Prove that 4(AQ² + BP²) = 5AB²

9. In a triangle ABC, AD is perpendicular on BC. Prove that AB² + CD² = AC² + BD²

10. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

11. In adjoining figure, OD, OE and OF are respectively perpendiculars to the sides BC, CA and AB from any point O in the interior of the triangle Prove that
(i) OA² + OB² + OC² – OD² – OE² – OF² = AF² + BD² + CE²
(ii) AF² + BD²+ CE² = AE² + CD² + BF²

12. O is any point in the insertor of a rectangle ABCD. Prove that interior OB² + OD² = OC² + OA²