1.   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

2.   PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. 
     Show that $PM^2 = QM . MR$.
	 
3.   In Fig., ABD is a triangle right angled at A and AC ⊥BD. 
	 

	Show that
	(i)  $AB^2 = BC . BD$ 
	(ii)  $AC^2 = BC . DC $
	(iii) $ AD^2 = BD . CD$

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

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

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

7.   Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of 
     its diagonals. 
	 
8.   In Fig. , O is a point in the interior of a triangle ABC, OD ⊥BC, OE ⊥AC and OF ⊥AB.
     
	 
	 Show that
	(i)  $OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2 =AF^2 + BD^2 + CE^2$
	(ii)  $AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2$.

9.   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.

10.   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?

11.   An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time,
     another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart 
     will be the two planes after $1{1/2}$ hours?
	 
12.   Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m,
	  find the distance between their tops.

13.   D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C.
      Prove that $AE^2 + BD^2 = AB^2 + DE^2$.

14.   The  perpendicular  from A on  side  BC  of  a  ∆ABC intersects BC at D such that DB = 3 CD
     (see Fig. 6.55). Prove that $2 AB^2 = 2 AC^2 + BC^2$.
	 
15.   In an equilateral triangle ABC, D is a point on side BC such that $BD = 1/3 BC$. Prove that
      $9AD^2 = 7 AB^2$.
	  
16.   In an equilateral triangle, prove that three times the square of one side is equal to four times 
      the square of one of its altitudes.
	  
17.   Tick the correct answer and justify : In ∆ABC,AB =AB = 6√3  cm,AC = 12 cm and BC = 6 cm.
	 The angle B is :
	(A)  120° (B)  60° (C)  90°  (D)  45°