# NCERT Solutions for Class 10 Maths Unit 6

## Triangles Class 10

### Unit 6 Triangles Exercise 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 Solutions

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted △ ABC.
In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).
The measures of the interior angles of a triangle in Euclidean space always add up to 180 degrees. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.

Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.

Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)

### Exercise 6.1 : Solutions of Questions on Page Number : 122

Q1 :

Fill in the blanks using correct word given in the brackets:-

(i) All circles are __________. (congruent, similar)

(ii) All squares are __________. (similar, congruent)

(iii) All __________ triangles are similar. (isosceles, equilateral)

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________. (equal, proportional)

(i) Similar

(ii) Similar

(iii) Equilateral

(iv) (a) Equal

(b) Proportional

Q2 :

Give two different examples of pair of

(i) Similar figures

(ii)Non-similar figures

(i) Two equilateral triangles with sides 1 cm and 2 cm

Two squares with sides 1 cm and 2 cm

(ii) Trapezium and square

Triangle and parallelogram

Q3 :

State whether the following quadrilaterals are similar or not:

### Exercise 6.2 : Solutions of Questions on Page Number : 128

Q1 :

In figure.6.17. (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

(i)

(ii)

Q2 :

E and F are points on the sides PQ and PR respectively of a ΔPQR. For each of the following cases, state whether EF || QR.

(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii)PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm

Q3 :

In the following figure, if LM || CB and LN || CD, prove that

Q4 :

In the following figure, DE || AC and DF || AE. Prove that

Q5 :

In the following figure, DE || OQ and DF || OR, show that EF || QR.

Q6 :

In the following figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

Q7 :

Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Q8 :

Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

Q9 :

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that

Q10 :

The diagonals of a quadrilateral ABCD intersect each other at the point O such thatShow that ABCD is a trapezium.

### Exercise 6.3 : Solutions of Questions on Page Number : 138

Q1 :

State which pairs of triangles in the following figure are similar? Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Q2 :

In the following figure, ΔODC ∝¼ ΔOBA, ∠ BOC = 125° and ∠ CDO = 70°. Find ∠ DOC, ∠ DCO and ∠ OAB

Q3 :

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that

Q4 :

In the following figure, Show that

Q5 :

S and T are point on sides PR and QR of ΔPQR such that ∠ P = ∠ RTS. Show that ΔRPQ ∠Â¼ ΔRTS.

Q6 :

In the following figure, if ΔABE ≅ ΔACD, show that ΔADE ∝¼ ΔABC.

Q7 :

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

(i) ΔAEP ∝¼ ΔCDP

(ii) ΔABD ∝¼ ΔCBE

(v) ΔPDC ∝¼ ΔBEC

Q8 :

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ∠Â¼ ΔCFB

Q9 :

In the following figure, ABC and AMP are two right triangles, right angled at B and M respectively, prove that:

(i) ΔABC Ã¢Ë†Â¼ ΔAMP

(ii)

Q10 :

CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ΔABC and ΔEFG respectively. If ΔABC Ã¢Ë†Â¼ ΔFEG, Show that:

(i)

(ii) ΔDCB Ã¢Ë†Â¼ ΔHGE

(iii) ΔDCA Ã¢Ë†Â¼ ΔHGF

Q11 :

In the following figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that ΔABD ∝¼ ΔECF

Q12 :

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∠Â¼ ΔPQR.

Q13 :

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that

Q14 :

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that

Q15 :

A vertical pole of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

Q16 :

If AD and PM are medians of triangles ABC and PQR, respectively where

### Exercise 6.4 : Solutions of Questions on Page Number : 143

Q1 :

Let and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.

Q2 :

Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD.

Q3 :

In the following figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that

Q4 :

If the areas of two similar triangles are equal, prove that they are congruent.

Q5 :

D, E and F are respectively the mid-points of sides AB, BC and CA of ΔABC. Find the ratio of the area of ΔDEF and ΔABC.

Q6 :

Prove that the ratio of the areas of two similar triangles is equal to the square

of the ratio of their corresponding medians.

Q7 :

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

Q8 :

ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the area of triangles ABC and BDE is

(A) 2 : 1

(B) 1 : 2

(C) 4 : 1

(D) 1 : 4

Q9 :

Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

(A) 2 : 3

(B) 4 : 9

(C) 81 : 16

(D) 16 : 81

### Exercise 6.5 : Solutions of Questions on Page Number : 150

Q1 :

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

Q2 :

PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM x MR.

Q3 :

ABC is an isosceles triangle right angled at C. prove that AB2 = 2 AC2.

Q4 :

ABC is an isosceles triangle with AC = BC. If AB2 = 2 AC2, prove that ABC is a right triangle.

Q5 :

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

Q6 :

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

Q7 :

In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that

(i) OA2 + OB2 + OC2 - OD2 - OE2 - OF2 = AF2 + BD2 + CE2

(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2

Q8 :

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.

Q9 :

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?

Q10 :

An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after hours?

Q11 :

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.

Q12 :

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2

Q13 :

The perpendicular from A on side BC of a ΔABC intersect BC at D such that DB = 3 CD. Prove that 2 AB2 = 2 AC2 + BC2

Q14 :

In an equilateral triangle ABC, D is a point on side BC such that BD = BC. Prove that 9 AD2 = 7 AB2.

Q15 :

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.

Q16 :

Tick the correct answer and justify: In ΔABC, AB = cm, AC = 12 cm and BC = 6 cm.

The angle B is:

(A) 120° (B) 60°

(C) 90° (D) 45°

### Exercise 6.6 : Solutions of Questions on Page Number : 152

Q1 :

In the given figure, PS is the bisector of ∠QPR of ΔPQR. Prove that .

Q2 :

In the given figure, D is a point on hypotenuse AC of ΔABC, DM ⊥ BC and DN ⊥ AB, Prove that:

(i) DM2 = DN.MC

(ii) DN2 = DM.AN

Q3 :

In the given figure, ABC is a triangle in which ∠ ABC> 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC.BD.

Q4 :

In the given figure, ABC is a triangle in which ∠ ABC < 90° and AD ⊥ BC. Prove that AC2 = AB2 + BC2 - 2BC.BD.

Q5 :

In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:

(i)

(ii)

(iii)