In \(AD\bot BC\) prove that AB² + CD² = BD² + AC². 

BL and CM are medians of a triangle ABC right angled at A.
Prove that 4(BL² + CM²) = 5BC².

Prove that in a right triangle, the square of 8. the hypotenure is equal to the sum of the squares of the others two sides.

A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.

In figure 0 is any point inside a rectangle ABCD. Prove that OB² + OD² = OA² + OC²

In \(\angle ACD={ 90 }^{ 0 }\) and \(CD\bot AB\) Prove that \(\cfrac { { BC }^{ 2 } }{ { AC }^{ 2 } } =\cfrac { BD }{ AD } \)

The perpendicular from A on side BC at a \(\triangle\)ABC intersects BC at D such that DB = 3 CD. Prove that 2AB² = 2AC² + BC².

In the figure \(\triangle O D C \sim \triangle O B A, \angle B O C=125^{\circ} \text { and }\angle C D O=70^{\circ} \text {, find } \angle D O C, \angle D C O \text { and } \angle O A B\)

\(\text { In the figure } \frac{Q R}{Q S}=\frac{Q T}{P R} \text { and } \angle 1=\angle 2 \text { show }\text { that } \triangle P Q S \sim \triangle T Q R\)

S and T are points on sides PR and QR of \(\triangle P Q R \text { such that } \angle P=\angle R T S\)
\(\text { Show that } \triangle R P Q \sim \triangle R T S\)

In the figure E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If \(\mathrm{AD}+\mathrm{BC} \text { and } \mathrm{EF} \perp \mathrm{AC} .\text { Prove that } \triangle A B D-\triangle E C F\)

The bisector of interior \(\angle\)A of \(\Delta\)ABC meets BC in D and the bisector of exterior \(\angle\)A meets BC produced in E. prove that \(\frac{B D}{B E}=\frac{C D}{C E}\)

If the diagonal BD of a quadrilateral ABCD bisects both \(\angle B \text { and } \angle D,\text { Show that } \frac{A B}{B C}=\frac{A D}{C D} \text {. }\)

A ladder 15 m long reaches a window which is 9 m above the ground on one side of a street. Keeping its foot at the same point, the ladder is trurned to other side of the street to reach a window 12 mhigh. Find the width of the street.

Prove that three times the square of any side of an eqlilateral triangle in equal to four times the square of the altitude.

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Two concentric circles have a common center 'O' the chord AB to the bigger ciicles touches the smaller circle at P. If OP = 3 cm and AB = 8 cm then find the radiu of the bigger circle.

Show that in a triangle, the medians are concurrent.