#### Chapters

Chapter 2 - Polynomials

Chapter 3 - Pair of Linear Equations in Two Variables

Chapter 4 - Triangles

Chapter 5 - Trigonometric Ratios

Chapter 6 - Trigonometric Identities

Chapter 7 - Statistics

Chapter 8 - Quadratic Equations

Chapter 9 - Arithmetic Progression

Chapter 10 - Circles

Chapter 11 - Constructions

Chapter 12 - Trigonometry

Chapter 13 - Probability

Chapter 14 - Co-Ordinate Geometry

Chapter 15 - Areas Related to Circles

Chapter 16 - Surface Areas and Volumes

## Chapter 4 - Triangles

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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)

Two triangles are similar, if their corresponding angles are .......... (proportional, equal)

Two triangles are similar, if their corresponding sides are .......... (proportional, equal)

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)

Write the truth value (T/F) of each of the following statement

Any two similar figures are congruent.

Write the truth value (T/F) of each of the following statement

Any two congruent figures are similar.

Write the truth value (T/F) of each of the following statement

Two polygons are similar, if their corresponding sides are proportional.

Write the truth value (T/F) of each of the following statement

Two polygons are similar if their corresponding angles are proportional.

Write the truth value (T/F) of each of the following statement

Two triangles are similar if their corresponding sides are proportional.

Write the truth value (T/F) of each of the following statement

Two triangles are similar if their corresponding angles are proportional.

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In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If `"AD"/"DB"=3/4` and AC = 15 cm, find AE

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If `"AD"/"DB"=3/4` and AC = 15 cm, find AE

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If `"AD"/"DB"=2/3` and AC = 18 cm, find AE

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If `"AD"/"DB"=2/3` and AC = 18 cm, find AE

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4, AE = 8, DB = x – 4, and EC = 3x – 19, find x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4, AE = 8, DB = x – 4, and EC = 3x – 19, find x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 8cm, AB = 12 cm and AE = 12 cm, find CE.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 8cm, AB = 12 cm and AE = 12 cm, find CE.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 2 cm, AB = 6 cm and AC = 9 cm, find AE.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 2 cm, AB = 6 cm and AC = 9 cm, find AE.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If `"AD"/"BD"=4/5` and EC = 2.5 cm, find AE

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If `"AD"/"BD"=4/5` and EC = 2.5 cm, find AE

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 8x − 7, DB = 5x − 3, AE = 4x − 3 and EC = (3x − 1), find the value of x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 8x − 7, DB = 5x − 3, AE = 4x − 3 and EC = (3x − 1), find the value of x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4x − 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x − 3, find the volume of x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4x − 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x − 3, find the volume of x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm, find the length of AC.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm, find the length of AC.

In a ΔABC, D and E are points on the sides AB and AC respectively. For the following case show that DE || BC

AB = 2cm, AD = 8cm, AE = 12 cm and AC = l8cm.

In a ΔABC, D and E are points on the sides AB and AC respectively. For the following case show that DE || BC

AB = 2cm, AD = 8cm, AE = 12 cm and AC = l8cm.

In a ΔABC, D and E are points on the sides AB and AC respectively. For the following case show that DE || BC

AB = 5.6cm, AD = 1.4cm, AC= 7.2 cm and AE = 1.8 cm.

AB = 5.6cm, AD = 1.4cm, AC= 7.2 cm and AE = 1.8 cm.

AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm.

AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm.

In a ΔABC, P and Q are points on sides AB and AC respectively, such that PQ || BC. If AP = 2.4 cm, AQ = 2 cm, QC = 3 cm and BC = 6 cm, find AB and PQ.

In a ΔABC, P and Q are points on sides AB and AC respectively, such that PQ || BC. If AP = 2.4 cm, AQ = 2 cm, QC = 3 cm and BC = 6 cm, find AB and PQ.

In a ΔABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 2.4cm, AE = 3.2 cm, DE = 2cm and BC = 5 cm, find BD and CE.

In a ΔABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 2.4cm, AE = 3.2 cm, DE = 2cm and BC = 5 cm, find BD and CE.

In below Fig., state if PQ || EF.

In below Fig., state if PQ || EF.

M and N are points on the sides PQ and PR respectively of a ΔPQR. For the following case, state whether MN || QR

PM = 4cm, QM = 4.5 cm, PN = 4 cm and NR = 4.5 cm

M and N are points on the sides PQ and PR respectively of a ΔPQR. For the following case, state whether MN || QR

PM = 4cm, QM = 4.5 cm, PN = 4 cm and NR = 4.5 cm

In three line segments OA, OB, and OC, points L, M, N respectively are so chosen that LM || AB and MN || BC but neither of L, M, N nor of A, B, C are collinear. Show that LN ||AC.

In three line segments OA, OB, and OC, points L, M, N respectively are so chosen that LM || AB and MN || BC but neither of L, M, N nor of A, B, C are collinear. Show that LN ||AC.

If D and E are points on sides AB and AC respectively of a ΔABC such that DE || BC and BD = CE. Prove that ΔABC is isosceles.

If D and E are points on sides AB and AC respectively of a ΔABC such that DE || BC and BD = CE. Prove that ΔABC is isosceles.

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In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If BD = 2.5cm, AB = 5cm and AC = 4.2cm, find DC.

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If BD = 2cm, AB = 5cm and DC = 3cm, find AC.

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If AB = 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD.

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If AB = 10 cm, AC =14 cm and BC =6 cm, find BD and DC.

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If AC = 4.2 cm, DC = 6 cm and 10 cm, find AB

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If AB = 5.6 cm, AC = 6cm and DC = 3cm, find BC.

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If AD = 5.6 cm, BC = 6cm and BD = 3.2 cm, find AC.

In a ΔABC, AD is the bisector of ∠A, meeting side BC at D.

If AB = 10cm, AC = 6 cm and BC = 12 cm, find BD and DC.

In the following Figure, AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB = 10cm, AC = 6cm and BC = 12 cm, find CE.

In the following Figure, ΔABC is a triangle such that `"AB"/"AC"="BD"/"DC"`, ∠B = 70°, ∠C = 50°. Find ∠BAD.

In ΔABC , if ∠1 = ∠2, prove that `"AB"/"AC"="BD"/"DC"`

D, E and F are the points on sides BC, CA and AB respectively of ΔABC such that AD bisects ∠A, BE bisects ∠B and CF bisects ∠C. If AB = 5 cm, BC = 8 cm and CA = 4 cm, determine AP, CE and BD.

In the following figure, check whether AD is the bisector of ∠A of ∆ABC in the following case:

AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm

In the following figure, check whether AD is the bisector of ∠A of ∆ABC in the following case:

AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm

In the following figure, check whether AD is the bisector of ∠A of ∆ABC in the following case:

AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm

In the following figure, check whether AD is the bisector of ∠A of ∆ABC in the following case:

AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm

In the following figure, check whether AD is the bisector of ∠A of ∆ABC in the following case:

AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm

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In below figure, If AB || CD, find the value of x.

In the below figure, If AB || CD, find the value of x.

In below figure, AB || CD. If OA = 3x – 19, OB = x – 4, OC = x – 3 and OD = 4, find x.

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

A vertical stick 10 cm long casts a shadow 8 cm long. At the same time a shadow 30 m long. Determine the height of the tower.

In the following figure, AB || QR. Find the length of PB.

In the following figure, XY || BC. Find the length of XY.

In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx.

In the following Figure, ∠ABC = 90° and BD ⊥ AC. If BD = 8 cm and AD = 4 cm, find CD.

In the following Figure, ∠ABC = 90° and BD ⊥ AC. If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.

In the following Figure, DE || BC such that AE = (1/4) AC. If AB = 6 cm, find AD.

In the following figure, PA, QB and RC are each perpendicular to AC. Prove that `1/x+1/z+1/y`

In below figure, ∠A = ∠CED, Prove that ΔCAB ~ ΔCED. Also, find the value of x.

The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of first triangle is 9 cm, what is the corresponding side of the other triangle?

In ΔABC and ΔDEF, it is being given that: AB = 5 cm, BC = 4 cm and CA = 4.2 cm; DE=10cm, EF = 8 cm and FD = 8.4 cm. If AL ⊥ BC and DM ⊥ EF, find AL: DM.

D and E are the points on the sides AB and AC respectively of a ΔABC such that: AD = 8 cm, DB = 12 cm, AE = 6 cm and CE = 9 cm. Prove that BC = 5/2 DE.

D and E are the points on the sides AB and AC respectively of a ΔABC such that: AD = 8 cm, DB = 12 cm, AE = 6 cm and CE = 9 cm. Prove that BC = 5/2 DE.

D is the mid-point of side BC of a ΔABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE : EX = 3 : 1

ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the AB and BC.

In ΔABC, AL and CM are the perpendiculars from the vertices A and C to BC and AB respectively. If AL and CM intersect at O, prove that:

(i) ΔOMA and ΔOLC

(ii) `"OA"/"OC"="OM"/"OL"`

In Fig below we have AB || CD || EF. If AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm, calculate the values of x and y.

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

In Figure below, if AB ⊥ BC, DC ⊥ BC and DE ⊥ AC, Prove that Δ CED ~ ABC.

Given: AB ⊥ BC, DC ⊥ BC and DE ⊥ AC

To prove: ΔCED ~ ΔABC

Proof:

∠BAC + ∠BCA = 90° …(i) [By angle sum property]

And, ∠BCA + ∠ECD = 90° …(ii) [DC ⊥ BC given]

Compare equation (i) and (ii)

∠BAC = ∠ECD …(iii)

In ΔCED and ΔABC

∠CED = ∠ABC [Each 90°]

∠ECD = ∠BAC [From (iii)]

Then, ΔCED ~ ΔABC [By AA similarity]

In an isosceles ΔABC, the base AB is produced both the ways to P and Q such that AP × BQ = AC^{2}. Prove that ΔAPC ~ ΔBCQ.

A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2m/sec. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using similarity criterion for two triangles, show that `"OA"/"OC"="OB"/"OD"`

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)`

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

In below Figure, ΔABC is right angled at C and DE ⊥ AB. Prove that ΔABC ~ ΔADE and Hence find the lengths of AE and DE.

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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.

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

Triangles ABC and DEF are similar If AC = 19cm and DF = 8 cm, find the ratio of the area of two triangles.

Triangles ABC and DEF are similar If area (ΔABC) = 36 cm^{2}, area (ΔDEF) = 64 cm^{2} and DE = 6.2 cm, find AB.

Triangles ABC and DEF are similar If AB = 1.2 cm and DE = 1.4 cm, find the ratio of the areas of ΔABC and ΔDEF.

In figure below ΔACB ~ ΔAPQ. If BC = 10 cm, PQ = 5 cm, BA = 6.5 cm and AP = 2.8 cm,

find CA and AQ. Also, find the area (ΔACB): area (ΔAPQ)

The areas of two similar triangles are 81 cm^{2} and 49 cm^{2} respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians?

The areas of two similar triangles are 169 cm^{2} and 121 cm^{2} respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle.

Two isosceles triangles have equal vertical angles and their areas are in the ratio 36 : 25. Find the ratio of their corresponding heights.

The areas of two similar triangles are 25 cm^{2} and 36 cm^{2} respectively. If the altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other.

The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

ABC is a triangle in which ∠A =90°, AN⊥ BC, BC = 12 cm and AC = 5cm. Find the ratio of the areas of ΔANC and ΔABC.

In Figure, DE || BC If DE = 4 cm, BC = 6 cm and Area (ΔADE) = 16 cm2, find the area of ΔABC.

In Figure, DE || BC If DE = 4cm, BC = 8 cm and Area (ΔADE) = 25 cm2, find the area of ΔABC.

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

In ΔABC, D and E are the mid-points of AB and AC respectively. Find the ratio of the areas of ΔADE and ΔABC

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)`

ABCD is a trapezium in which AB || CD. The diagonals AC and BD intersect at O. Prove that: (i) ΔAOB and ΔCOD (ii) If OA = 6 cm, OC = 8 cm,

Find:(a) `("area"(triangleAOB))/("area"(triangleCOD))`

(b) `("area"(triangleAOD))/("area"(triangleCOD))`

In ABC, P divides the side AB such that AP : PB = 1 : 2. Q is a point in AC such that PQ || BC. Find the ratio of the areas of ΔAPQ and trapezium BPQC.

The areas of two similar triangles are 100 cm^{2} and 49 cm^{2} respectively. If the altitude the bigger triangle is 5 cm, find the corresponding altitude of the other.

The areas of two similar triangles are 121 cm^{2} and 64 cm^{2} respectively. If the median of the first triangle is 12.1 cm, find the corresponding median of the other.

If ΔABC ~ ΔDEF such that AB = 5 cm, area (ΔABC) = 20 cm^{2} and area (ΔDEF) = 45 cm^{2}, determine DE.

In ΔABC, PQ is a line segment intersecting AB at P and AC at Q such that PQ || BC and PQ divides ΔABC into two parts equal in area. Find `"BP"/"AB"`

The areas of two similar triangles ABC and PQR are in the ratio 9:16. If BC = 4.5 cm, find the length of QR.

ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 m, prove that area of ΔAPQ is one- sixteenth of the area of ABC.

If D is a point on the side AB of ΔABC such that AD : DB = 3.2 and E is a Point on BC such that DE || AC. Find the ratio of areas of ΔABC and ΔBDE.

If ΔABC and ΔBDE are equilateral triangles, where D is the mid-point of BC, find the ratio of areas of ΔABC and ΔBDE.

AD is an altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is constructed. Prove that Area (ΔADE): Area (ΔABC) = 3: 4

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If the sides of a triangle are 3 cm, 4 cm, and 6 cm long, determine whether the triangle is a right-angled triangle.

If the sides of a triangle are 3 cm, 4 cm, and 6 cm long, determine whether the triangle is a right-angled triangle.

The sides of triangle is given below. Determine it is right triangle or not.

a = 7 cm, b = 24 cm and c = 25 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 7 cm, b = 24 cm and c = 25 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 9 cm, b = l6 cm and c = 18 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 9 cm, b = l6 cm and c = 18 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 1.6 cm, b = 3.8 cm and c = 4 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 1.6 cm, b = 3.8 cm and c = 4 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 8 cm, b = 10 cm and c = 6 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 8 cm, b = 10 cm and c = 6 cm

A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?

A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?

A ladder 17 m long reaches a window of a building 15 m above the ground. Find the distance of the foot of the ladder from the building.

A ladder 17 m long reaches a window of a building 15 m above the ground. Find the distance of the foot of the ladder from the building.

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.

In an isosceles triangle ABC, AB = AC = 25 cm, BC = 14 cm. Calculate the altitude from A on BC.

In an isosceles triangle ABC, AB = AC = 25 cm, BC = 14 cm. Calculate the altitude from A on BC.

The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its tip reach?

The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its tip reach?

Two poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

Two poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

Using Pythagoras theorem determine the length of AD in terms of b and c shown in Figure.

Using Pythagoras theorem determine the length of AD in terms of b and c shown in Figure.

A triangle has sides 5 cm, 12 cm and 13 cm. Find the length to one decimal place, of the perpendicular from the opposite vertex to the side whose length is 13 cm.

A triangle has sides 5 cm, 12 cm and 13 cm. Find the length to one decimal place, of the perpendicular from the opposite vertex to the side whose length is 13 cm.

ABCD is a square. F is the mid-point of AB. BE is one third of BC. If the area of ΔFBE = 108 cm^{2}, find the length of AC.

ABCD is a square. F is the mid-point of AB. BE is one third of BC. If the area of ΔFBE = 108 cm^{2}, find the length of AC.

In an isosceles triangle ABC, if AB = AC = 13 cm and the altitude from A on BC is 5 cm, find BC.

In an isosceles triangle ABC, if AB = AC = 13 cm and the altitude from A on BC is 5 cm, find BC.

In a ΔABC, AB = BC = CA = 2a and AD ⊥ BC. Prove that

(i) AD = a`sqrt3`

(ii) Area (ΔABC) = `sqrt3` a^{2}

In a ΔABC, AB = BC = CA = 2a and AD ⊥ BC. Prove that

(i) AD = a`sqrt3`

(ii) Area (ΔABC) = `sqrt3` a^{2}

The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.

The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.

Each side of a rhombus is 10 cm. If one of its diagonals is 16 cm find the length of the other diagonal.

Each side of a rhombus is 10 cm. If one of its diagonals is 16 cm find the length of the other diagonal.

In an acute-angled triangle, express a median in terms of its sides.

In an acute-angled triangle, express a median in terms of its sides.

Calculate the height of an equilateral triangle each of whose sides measures 12 cm.

Calculate the height of an equilateral triangle each of whose sides measures 12 cm.

In right-angled triangle ABC in which ∠C = 90°, if D is the mid-point of BC, prove that AB^{2} = 4AD^{2} − 3AC^{2}.

In right-angled triangle ABC in which ∠C = 90°, if D is the mid-point of BC, prove that AB^{2} = 4AD^{2} − 3AC^{2}.

In Figure, D is the mid-point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED

= x, AD = p and AE = h, prove that:

(i) `b^2 = p^2 + ax + a^2/4`

(ii) `c^2 = p^2 - ax + a^2/4`

(iii) `b^2 + c^2 = 2p^2 + a^2/2`

In Figure, D is the mid-point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED

= x, AD = p and AE = h, prove that:

(i) `b^2 = p^2 + ax + a^2/4`

(ii) `c^2 = p^2 - ax + a^2/4`

(iii) `b^2 + c^2 = 2p^2 + a^2/2`

In the given figure, ∠B < 90° and segment AD ⊥ BC, show that

(i) b^{2 }= h^{2 }+ a^{2 }+ x^{2 }- 2ax

(ii) b^{2} = a^{2} + c^{2} - 2ax

In the given figure, ∠B < 90° and segment AD ⊥ BC, show that

(i) b^{2 }= h^{2 }+ a^{2 }+ x^{2 }- 2ax

(ii) b^{2} = a^{2} + c^{2} - 2ax

In ∆ABC, ∠A is obtuse, PB ⊥ AC and QC ⊥ AB. Prove that:

(i) AB ✕ AQ = AC ✕ AP

(ii) BC^{2} = (AC ✕ CP + AB ✕ BQ)

In ∆ABC, ∠A is obtuse, PB ⊥ AC and QC ⊥ AB. Prove that:

(i) AB ✕ AQ = AC ✕ AP

(ii) BC^{2} = (AC ✕ CP + AB ✕ BQ)

In a right ∆ABC right-angled at C, if D is the mid-point of BC, prove that BC^{2} = 4(AD^{2} − AC^{2}).

In a right ∆ABC right-angled at C, if D is the mid-point of BC, prove that BC^{2} = 4(AD^{2} − AC^{2}).

In a quadrilateral ABCD, ∠B = 90°, AD^{2} = AB^{2} + BC^{2} + CD^{2}, prove that ∠ACD = 90°.

In a quadrilateral ABCD, ∠B = 90°, AD^{2} = AB^{2} + BC^{2} + CD^{2}, prove that ∠ACD = 90°.

In an equilateral ΔABC, AD ⊥ BC, prove that AD^{2} = 3BD^{2}.

In an equilateral ΔABC, AD ⊥ BC, prove that AD^{2} = 3BD^{2}.

∆ABD is a right triangle right-angled at A and AC ⊥ BD. Show that

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

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

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

(iv) `"AB"^2/"AC"^2="BD"/"DC"`

∆ABD is a right triangle right-angled at A and AC ⊥ BD. Show that

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

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

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

(iv) `"AB"^2/"AC"^2="BD"/"DC"`

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 1000km/hr. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km/hr. How far apart will be the two planes after 1 hours?

An aeroplane leaves an airport and flies due north at a speed of 1000km/hr. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km/hr. How far apart will be the two planes after 1 hours?

Determine whether the triangle having sides (a − 1) cm, 2`sqrta` cm and (a + 1) cm is a right-angled

triangle.

Determine whether the triangle having sides (a − 1) cm, 2`sqrta` cm and (a + 1) cm is a right-angled

triangle.

#### Extra questions

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = x, DB = x − 2, AE = x + 2 and EC = x − 1, find the value of x.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = x, DB = x − 2, AE = x + 2 and EC = x − 1, find the value of x.

#### Textbook solutions for Class 10

## R.D. Sharma solutions for Class 10 Mathematics chapter 4 - Triangles

R.D. Sharma solutions for Class 10 Mathematics chapter 4 (Triangles) include all questions with solution and detail explanation from 10 Mathematics. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 4 Triangles are Converse of Basic Proportionality Theorem, Property of three parallel lines and their transversals, Property of an Angle Bisector of a Triangle, Basic Proportionality Theorem Or Thales Theorem, Appolonius Theorem, Application of Pythagoras Theorem in Acute Angle and Obtuse Angle, Introduction to Similarity, Properties of Ratios of Areas of Two Triangles, Similarity of Triangles, Similar Triangles, Similarity Triangle Theorem, Areas of Two Similar Triangles, Similar Figures, Criteria for Similarity of Triangles, Pythagoras Theorem, Areas of Similar Triangles, Similarity of Triangles, Similarity Triangle Theorem, Application of Pythagoras Theorem in Acute Angle and Obtuse Angle, Triangles Examples and Solutions, Basic Proportionality Theorem Or Thales Theorem, Angle Bisector, Similarity Examples and Solutions, Ratio of Sides of Triangle.

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