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# NCERT solutions Mathematics Class 10 chapter 6 Triangles

## Chapter 6 - Triangles

#### Page 122

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

All circles are __________. (congruent, similar)

Q 1.1 | Page 122

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

All squares are __________. (similar, congruent)

Q 1.2 | Page 122

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

All __________ triangles are similar. (isosceles, equilateral)

Q 1.3 | Page 122

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

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

Q 1.4 | Page 122

Give two different examples of pair of similar figures

Q 2.1 | Page 122

Give two different examples of pair of Non-similar figures

Q 2.1 | Page 122

State whether the following quadrilaterals are similar or not:

Q 3 | Page 122

#### Pages 128 - 129

See the given Figure.  DE || BC. Find EC

Q 1.1 | Page 128

See the given Figure.  DE || BC. Find AD

Q 1.2 | Page 128

E and F are points on the sides PQ and PR respectively of a ΔPQR. For the following case, state whether EF || QR. PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

Q 2.1 | Page 128

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

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

Q 2.2 | Page 128

E and F are points on the sides PQ and PR respectively of a ΔPQR. For the following case, state whether EF || QR. PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

Q 2.3 | Page 128

In the following figure, if LM || CB and LN || CD, prove that (AM)/(AB)=(AN)/(AD)

Q 3 | Page 128

In the following figure, DE || AC and DF || AE. Prove that (BF)/(FE)=(BE)/(EC)

Q 4 | Page 128

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

Q 5 | Page 129

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.

Q 6 | Page 129

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).

Q 7 | Page 129

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).

Q 8 | Page 129

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that (AO)/(BO) = (CO)/(DO)

Q 9 | Page 129

The diagonals of a quadrilateral ABCD intersect each other at the point O such that (AO)/(BO) = (CO)/(DO)  Show that ABCD is a trapezium

Q 10 | Page 129

#### Pages 139 - 141

State which pair 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:

Q 1.1 | Page 139

State which pair of triangles in the given 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:

Q 1.2 | Page 139

State which pair 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:

Q 1.3 | Page 139

State which pair 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:

Q 1.4 | Page 139

State which pair 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:

Q 1.5 | Page 139

State which pair 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:

Q 1.6 | Page 139

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

Q 2 | Page 139

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 (AO)/(OC) = (OB)/(OD)

Q 3 | Page 139

In the following figure, (QR)/(QS) = (QT)/(PR)  and ∠1 = ∠2. Show that ΔPQS ~ ΔTQR.

Q 4 | Page 140

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

Q 5 | Page 140

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

Q 6 | Page 140

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

ΔAEP ∼ ΔCDP

Q 7.1 | Page 140

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

ΔABD ∼ ΔCBE

Q 7.2 | Page 140

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

Q 7.3 | Page 140

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

ΔPDC ∼ ΔBEC

Q 7.4 | Page 140

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

Q 8 | Page 140

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

ΔABC ~ ΔAMP

 (CA)/(PA) = (BC)/(MP)

Q 9 | Page 140

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 ~ ΔFEG, Show that

(i) (CD)/(GH) = (AC)/(FG)

(ii) ΔDCB ~ ΔHGE

(iii) ΔDCA ~ ΔHGF

Q 10 | Page 140

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 ∼ ΔECF

Q 11 | Page 141

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 ∼ ΔPQR.

Q 12 | Page 141

D is a point on the side BC of ∆ABC such that ∠ADC = ∠BAC. Prove that  \frac{CA}{CD}=\frac{CB}{CA} or, CA^2 = CB × CD.

Q 13 | Page 141

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 ΔABC ~ ΔPQR

Q 14 | Page 141

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.

Q 15 | Page 141

If AD and PM are medians of triangles ABC and PQR, respectively where Δ ABC ~ Δ PQR, prove that (AB)/(PQ) = (AD)/(PM)

Q 16 | Page 141

#### Page 143

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

Q 1 | Page 143

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.

Q 2 | Page 143

In Figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that (ar(ABC))/(ar(DBC)) = (AO)/(DO)

Q 3 | Page 143

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

Q 4 | Page 143

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.

Q 5 | Page 143

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

Q 6 | Page 143

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

Q 7 | Page 143

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

Q 8 | Page 143

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

Q 9 | Page 143

#### Pages 150 - 151

Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 7 cm, 24 cm, 25 cm

Q 1.1 | Page 150

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. 3 cm, 8 cm, 6 cm

Q 1.2 | Page 150

Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 50 cm, 80 cm, 100 cm

Q 1.3 | Page 150

Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 13 cm, 12 cm, 5 cm

Q 1.4 | Page 150

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

Q 2 | Page 150

In Figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that AB2 = BC × BD

Q 3.1 | Page 150

In Figure ABD is a triangle right angled at A and AC ⊥ BD. Show that AC2 = BC × DC

Q 3.2 | Page 150

In Figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that AD2 = BD × CD

Q 3.3 | Page 150

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

Q 4 | Page 150

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

Q 5 | Page 150

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

Q 6 | Page 150

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

Q 7 | Page 150

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 + CE= AE2 + CD2 + BF2

Q 8 | Page 151

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.

Q 9 | Page 151

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?

Q 10 | Page 151

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 1 1/2 hours?

Q 11 | Page 151

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.

Q 12 | Page 151

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

Q 13 | Page 151

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

Q 14 | Page 151

In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3BC . Prove that 9 AD2 = 7 AB2

Q 15 | Page 151

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.

Q 16 | Page 151

Tick the correct answer and justify: In ΔABC, AB = 6sqrt3 cm, AC = 12 cm and BC = 6 cm.

The angle B is:

(A) 120° (B) 60°

(C) 90° (D) 45°

Q 17 | Page 151

#### Pages 152 - 153

In the given figure, PS is the bisector of ∠QPR of ΔPQR. Prove that (QS)/(SR) = (PQ)/(PR)

Q 1 | Page 152

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

Q 3.1 | Page 152

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

Q 4 | Page 152

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

1)AC^2 = AD^2 + BC . DM + ((BC)/2)^2

2) AB^2 = AD^2 – BC . DM + ((BC)/2)^2

3) AC^2 + AB^2 = 2 AD^2 + 1/2(BC)^2

Q 5 | Page 152

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

Q 6 | Page 153

In the given figure, two chords AB and CD intersect each other at the point P. prove that:

(i) ΔAPC ∼ ΔDPB

(ii) AP.BP = CP.DP

Q 7 | Page 153

In the given figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that

(i) ΔPAC ∼ ΔPDB

(ii) PA.PB = PC.PD

Q 8 | Page 153

In the given figure, D is a point on side BC of ΔABC such that ∠ADC=∠BAC . Prove that AD is the bisector of ∠BAC.

Q 9 | Page 153

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, ho much string does she have out (see Figure)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

Q 10 | Page 153

#### Extra questions

In the given figure, DE || BC and DE : BC = 3 : 5. Calculate the ratio of the areas of ∆ADE and the trapezium BCED

From a point O in the interior of a ∆ABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove
that :

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

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

In Fig., ∆ABC is an obtuse triangle, obtuse angled at B. If AD ⊥ CB, prove that AC2 = AB2 + BC2 + 2BC × BD

In figure, \frac{AO}{OC}=\frac{BO}{OD}=\frac{1}{2} and AB = 5 cm. Find the value of DC.

P and Q are points on sides AB and AC respectively of ∆ABC. If AP = 3 cm, PB = 6cm. AQ = 5 cm and QC = 10 cm, show that BC = 3PQ.

ABCD is a rhombus. Prove that AB2 + BC2 + CD2 + DA2= AC2 + BD2

In a right triangle ABC right-angled at C, P and Q are the points on the sides CA and CB respectively, which divide these sides in the ratio 2 : 1. Prove that

(i) 9 AQ^2 = 9 AC^2 + 4 BC^2

(ii) 9 BP^2 = 9 BC^2 + 4 AC^2

(iii) 9 (AQ^2 + BP^2 ) = 13 AB^2

In figure, find ∠L

D, E, F are the mid-point of the sides BC, CA and AB respectively of a ∆ABC. Determine the ratio of the areas of ∆DEF and ∆ABC.

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

The perpendicular AD on the base BC of a ∆ABC intersects BC at D so that DB = 3 CD. Prove that 2AB^2 = 2AC^2 + BC^2

If ∆ABC ~ ∆DEF such that area of ∆ABC is 16cm2 and the area of ∆DEF is 25cm2 and BC = 2.3 cm. Find the length of EF.

In figure, ∆ACB ~ ∆APQ. If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm, AP = 2.8 cm, find CA and AQ.

In a trapezium ABCD, O is the point of intersection of AC and BD, AB || CD and AB = 2 × CD. If the area of ∆AOB = 84 cm2 . Find the area of ∆COD

P and Q are the mid-points of the sides CA and CB respectively of a ∆ABC, right angled at C. Prove that:

(i) 4AQ^2 = 4AC^2 + BC^2

(ii) 4BP^2 = 4BC^2 + AC^2

(iii) (4AQ^2 + BP^2 ) = 5AB^2

If ∆ABC is similar to ∆DEF such that ∆DEF = 64 cm2 , DE = 5.1 cm and area of ∆ABC = 9 cm2 . Determine the area of AB

Two towers of heights 10 m and 30 m stand on a plane ground. If the distance between their feet is 15 m, find the distance between their tops

ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB, prove that

(i) cp = ab

(ii) 1/p^2=1/a^2+1/b^2

Side of a triangle is given, determine it is a right triangle.

(2a – 1) cm, 2\sqrt { 2a } cm, and (2a + 1) cm

Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights

In two similar triangles ABC and PQR, if their corresponding altitudes AD and PS are in the ratio 4 : 9, find the ratio of the areas of ∆ABC and ∆PQR

Prove that the area of the triangle BCE described on one side BC of a square ABCD as base is one half the area of the similar Triangle ACF described on the diagonal AC as base

The areas of two similar triangles ∆ABC and ∆PQR are 25 cm2 and 49 cm2 respectively. If QR = 9.8 cm, find BC

D and E are points on the sides AB and AC respectively of a ∆ABC such that DE || BC and divides ∆ABC into two parts, equal in area. Find

In figure, considering triangles BEP and CPD, prove that BP × PD = EP × PC.

In figure, QA and PB are perpendicular to AB. If AO = 10 cm, BO = 6 cm and PB = 9 cm. Find AQ

In the figure, E is a point on side CB produced of an isosceles ∆ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that ∆ABD ~ ∆ECF

In figure, ∠CAB = 90º and AD ⊥ BC. If AC = 75 cm, AB = 1 m and BD = 1.25 m, find AD.

A vertical stick 20 cm long casts a shadow 6 cm long on the ground. At the same time, a tower casts a shadow 15 m long on the ground. Find the height of the tower.

In figure, ∠BAC = 90º and segment AD ⊥ BC. Prove that AD2 = BD × DC.

In figure, ∠A = ∠CED, prove that ∆CAB ~ ∆CED. Also, find the value of x.

In figure, if ∠A = ∠C, then prove that ∆AOB ~ ∆COD

In figure, ABCD is a trapezium with AB || DC. If ∆AED is similar to ∆BEC, prove that AD = BC.

A man goes 10 m due east and then 24 m due north. Find the distance from the starting point

Prove that the line segments joining the mid points of the sides of a triangle form four triangles, each of which is similar to the original triangle

The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB

In figure, ∠B of ∆ABC is an acute angle and AD ⊥ BC, prove that AC2 = AB2 + BC2 – 2BC × BD

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL = 2 BL

If ABC is an equilateral triangle of side a, prove that its altitude =  \frac { \sqrt { 3 } }{ 2 } a

Examine each pair of triangles in Figure, and state which pair of triangles are similar. Also, state the similarity criterion used by you for answering the question and write the similarity relation in symbolic form

figure (i)

figure 2

figure 3

figure 4

figure 5

figure 6

figure 7

In a ∆ABC, AD ⊥ BC and AD2 = BC × CD. Prove ∆ABC is a right triangle

If a perpendicular is drawn from the vertex containing the right angle of a right triangle to the hypotenuse then prove that the triangle on each side of the perpendicular are similar to each other and to the original triangle. Also, prove that the square of the perpendicular is equal to the product of the lengths of the two parts of the hypotenuse

The diagonal BD of a parallelogram ABCD intersects the segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA

ABC is a right-angled triangle, right-angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 5 cm and 12 cm. Find the radius of the circle

In ∆ABC, DE is parallel to base BC, with D on AB and E on AC. If \frac{AD}{DB}=\frac{2}{3} , find \frac{BC}{DE}.

In an isosceles ∆ABC, the base AB is produced both ways in P and Q such that AP × BQ = AC2 and CE are the altitudes. Prove that ∆ACP ~ ∆BCQ.

S