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#### Chapters

1 - Real Numbers2 - Polynomials

3 - Pair of Linear Equations in Two Variables

4 - Quadratic Equations

5 - Arithmetic Progressions

6 - Triangles

7 - Coordinate Geometry

8 - Introduction to Trigonometry

9 - Some Applications of Trigonometry

10 - Circles

11 - Constructions

12 - Areas Related to Circles

13 - Surface Areas and Volumes

14 - Statistics

15 - Probability

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### Chapter 6 - Triangles

#### Page 122

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

All circles are __________. (congruent, similar)

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

All squares are __________. (similar, congruent)

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

All __________ triangles are similar. (isosceles, equilateral)

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)

Give two different examples of pair of Non-similar figures

Give two different examples of pair of similar figures

State whether the following quadrilaterals are similar or not:

#### Pages 128 - 129

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

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

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

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

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

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

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

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

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.

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).

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).

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

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

#### 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:

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:

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:

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:

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

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)`

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

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

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

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

ΔAEP ∼ ΔCDP

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

ΔABD ∼ ΔCBE

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

ΔAEP ∼ ΔADB

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

ΔPDC ∼ ΔBEC

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

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)`

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

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

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.

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.`

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

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.

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

#### Page 143

Let Δ ABC ~ Δ DEF and their areas be, respectively, 64 cm^{2} and 121 cm^{2}. If EF = 15.4 cm, find BC

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.

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)`

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

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.

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

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

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

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

#### 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

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

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

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

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

In Figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that AB^{2} = BC × BD

In Figure ABD is a triangle right angled at A and AC ⊥ BD. Show that AC^{2} = BC × DC

In Figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that AD^{2} = BD × CD

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

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

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

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

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) 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}

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.

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?

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?

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.

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}

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

In an equilateral triangle ABC, D is a point on side BC such that BD = `1/3BC` . Prove that 9 AD^{2} = 7 AB^{2}

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.

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°

#### Pages 152 - 153

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

In the given figure, ABC is a triangle in which ∠ABC> 90° and AD ⊥ CB produced. Prove that AC^{2} = AB^{2} + BC^{2} + 2BC.BD.

In the given figure, ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC^{2} = AB^{2} + BC^{2} − 2BC.BD.

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`

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

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

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

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.

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?

#### Extra questions

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

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

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

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

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

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

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

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

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

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

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 areas of two similar triangles ∆ABC and ∆PQR are 25 cm^{2} and 49 cm^{2} respectively. If QR = 9.8 cm, find BC

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

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`

If ∆ABC is similar to ∆DEF such that ∆DEF = 64 cm^{2} , DE = 5.1 cm and area of ∆ABC = 9 cm^{2} . Determine the area of AB

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`

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`

If ∆ABC ~ ∆DEF such that area of ∆ABC is 16cm^{2} and the area of ∆DEF is 25cm^{2} and BC = 2.3 cm. Find the length of EF.

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 cm^{2} . Find the area of ∆COD

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

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.

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

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

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

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.

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 figure, ∠B of ∆ABC is an acute angle and AD ⊥ BC, prove that AC^{2} = AB^{2} + BC^{2} – 2BC × BD

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

In figure, find ∠L

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`

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

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`

In a ∆ABC, AD ⊥ BC and AD^{2} = BC × CD. Prove ∆ABC is a right triangle

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

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

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.

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, ∠BAC = 90º and segment AD ⊥ BC. Prove that AD^{2} = BD × DC.

ABCD is a rhombus. Prove that AB^{2} + BC^{2} + CD^{2} + DA^{2}= AC^{2} + BD^{2}

In Fig., ∆ABC is an obtuse triangle, obtuse angled at B. If AD ⊥ CB, prove that AC^{2} = AB^{2} + BC^{2} + 2BC × BD

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

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

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

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

In an isosceles ∆ABC, the base AB is produced both ways in P and Q such that AP × BQ = AC^{2} and CE are the altitudes. Prove that ∆ACP ~ ∆BCQ.

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

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