**Ex 6.5 Class 10 Maths Question 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

**Solution for Ex 6.5 Class 10 Maths Question 1**

**Solution for Ex 6.5 Class 10 Maths Question 1**

**Ex 6.5 Class 10 Maths Question 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 X MR.

**Solution for Ex 6.5 Class 10 Maths Question 2**

**Solution for Ex 6.5 Class 10 Maths Question 2**

**Ex 6.5 Class 10 Maths Question 3.**

In the given figure, ABD is a triangle right angled at A and AC i. BD. Show that

(i) AB^{2} = BC.BD

(ii) AC^{2} = BC.DC

(iii) AD^{2} = BD.CD

**Solution for Ex 6.5 Class 10 Maths Question 3**

**Solution for Ex 6.5 Class 10 Maths Question 3**

**Ex 6.5 Class 10 Maths Question 4.**

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

**Solution for Ex 6.5 Class 10 Maths Question 4**

**Solution for Ex 6.5 Class 10 Maths Question 4**

**Ex 6.5 Class 10 Maths Question 5.**

ABC is an isosceles triangle with AC = BC. If AB^{2} = 2AC^{2} , Prove that ABC is a right triangle.

**Solution for Ex 6.5 Class 10 Maths Question 5**

**Solution for Ex 6.5 Class 10 Maths Question 5**

**Ex 6.5 Class 10 Maths Question 6.**

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

**Solution for Ex 6.5 Class 10 Maths Question 6**

**Solution for Ex 6.5 Class 10 Maths Question 6**

**Ex 6.5 Class 10 Maths Question 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.

**Solution for Ex 6.5 Class 10 Maths Question 7**

**Solution for Ex 6.5 Class 10 Maths Question 7**

**Ex 6.5 Class 10 Maths Question 8.**

In the given figure, 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}.

**Solution for Ex 6.5 Class 10 Maths Question 8**

**Solution for Ex 6.5 Class 10 Maths Question 8**

**Ex 6.5 Class 10 Maths Question 9.**

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

**Solution for Ex 6.5 Class 10 Maths Question 9**

**Solution for Ex 6.5 Class 10 Maths Question 9**

**Ex 6.5 Class 10 Maths Question 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?

**Solution for Ex 6.5 Class 10 Maths Question 10**

**Solution for Ex 6.5 Class 10 Maths Question 10**

**Ex 6.5 Class 10 Maths Question 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 hours?

**Solution for Ex 6.5 Class 10 Maths Question 11**

**Solution for Ex 6.5 Class 10 Maths Question 11**

**Ex 6.5 Class 10 Maths Question 12.**

Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.

**Solution for Ex 6.5 Class 10 Maths Question 12**

**Solution for Ex 6.5 Class 10 Maths Question 12**

**Ex 6.5 Class 10 Maths Question 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}.

**Solution for Ex 6.5 Class 10 Maths Question 13**

**Solution for Ex 6.5 Class 10 Maths Question 13**

**Ex 6.5 Class 10 Maths Question 14.**

The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3CD (see the figure). Prove that 2AB^{2 }= 2AC^{2} + BC^{2}.

**Solution for Ex 6.5 Class 10 Maths Question 14**

**Solution for Ex 6.5 Class 10 Maths Question 14**

**Ex 6.5 Class 10 Maths Question 15.**

In an equilateral triangle ABC, D is a point on side BC, such that BD = BC. Prove that 9AD^{2} = 7AB^{2}.

**Solution for Ex 6.5 Class 10 Maths Question 15**

**Solution for Ex 6.5 Class 10 Maths Question 15**

**Ex 6.5 Class 10 Maths Question 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.

**Solution for Ex 6.5 Class 10 Maths Question 16**

**Solution for Ex 6.5 Class 10 Maths Question 16**

**Ex 6.5 Class 10 Maths Question 17.**

Tick the correct answer and justify : In ∆ABC, AB = 12 cm and BC = 6 cm. The angle B is:

(a) 120°

(b) 60°

(c) 90°

(d) 45

**Solution for Ex 6.5 Class 10 Maths Question 17**

**Solution for Ex 6.5 Class 10 Maths Question 17**