Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.

(i) 7 cm, 24 cm, 25 cm

(ii) 3 cm, 8 cm, 6 cm

(iii) 50 cm, 80 cm, 100 cm

(iv) 13 cm, 12 cm, 5 cm

PQR is a triangle right angled at P and M is a point on QR such that PM $\perp$ QR. Show that $PM^2$ = QM . MR.

In Fig. 6.53, ABD is a triangle right angled at A and AC $\perp$ BD. Show that

(i) $AB^2$ = BC . BD

(ii) $AC^2$ = BC . DC

(iii) $AD^2$ = BD . CD

ABC is an isosceles triangle right angled at C. Prove that $AB^2 = 2AC^2$

ABC is an isosceles triangle with AC = BC. If $AB^2 = 2 AC^2$, prove that ABC is a right triangle.

ABC is an equilateral triangle of side 2a. Find each of its altitudes.

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

In Fig. 6.54, O is a point in the interior of a triangle ABC, OD $\perp$ BC, OE $\perp$ AC and OF $\perp$ AB. Show that

(i) $OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2 = AF^2 + BD^2 + CE^2$

(ii) $AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2$

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?