Class 10

### Two poles of heights 6m and 11m stand vertically on a plane ground. If the distance between their feet is 12m, find the distance between their tops. (a) 12m (b) 13m (c) 14m (d) 15m

Correct Answer: (b)13 m Explanation:         Let the poles be and CD AB = 6 m CD = 11 m Let AC be 12 m Draw a perpendicular from CD, meeting CD at E. BE = 12 m Applying...

### The areas of two similar triangles are 25 and 36 respectively. If the altitude of the first triangle is 3.5cm, then the corresponding altitude of the other triangle. (a) 5.6cm (b) 6.3cm (c) 4.2cm (d) 7cm

Correct Answer: (c) 4.2cm Explanation: The ratio of areas of similar triangles is equal to the ratio of squares of their corresponding altitudes. Let ɦ be the altitude of the other triangle....

### If ∆ABC~∆DEF such that 2AB = DE and BC = 6cm, find EF.

Answer: ∆ABC ~ ∆ DEF $\begin{array}{l} \frac{{AB}}{{DE}} = \frac{{BC}}{{EF}}\\ \frac{1}{2} = \frac{6}{{EF}}\\ EF = 12cm \end{array}$

### In the given figure, DE║BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.

Answer:       DE || BC $\begin{array}{l} \frac{{AD}}{{DB}} = \frac{{AE}}{{EC}}\\ \frac{x}{3x+4} = \frac{x+3}{{3x+19}}\\ \end{array}$ ???? (3???? + 19) = (???? + 3)(3???? + 4) 3????2 +...

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

Answer:       Let the ladder be AB and BC be the height of the window from the ground. Given, AB 10 m BC = 8 m Applying theorem in right-angled triangle ACB, ????????2 = ????????2 +...

### The line segments joining the midpoints of the sides of a triangle form four triangles, each of which is (a) congruent to the original triangle (b) similar to the original triangle (c) an isosceles triangle (d) an equilateral triangle

Correct Answer: (b) similar to the original triangle Explanation:             The line segments joining the midpoint of the sides of a triangle form four triangles,...

### Two isosceles triangles have their corresponding angles equal and their areas are in the ratio 25: 36. The ratio of their corresponding heights is (a) 25 : 36 (b) 36 : 25 (c) 5 : 6 (d) 6: 5

Correct Answer: (c) 5:6 Explanation: Let x and y be the corresponding heights of the two triangles. The corresponding angles of the triangles are equal. The triangles are similar. (By AA criterion)...

### In an equilateral ∆ABC, D is the midpoint of AB and E is the midpoint of AC. Then, ar(∆ABC) : ar(∆ADE) = ? (a) 2 : 1 (b) 4 : 1 (c) 1 : 2 (d) 1 : 4

Correct Answer: (b) 4:1 Explanation:           In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. By midpoint theorem and Basic Proportionality Theorem,...

### It is given that ∆ABC~∆PQR and , then

Correct Answer: (d)\frac{9}{4}$Explanation: Given, ∆ABC ~ ∆PQR$\begin{array}{l} \frac{{BC}}{{QR}} = \frac{2}{3}\\ \end{array}\begin{array}{l} \frac{{ar(\Delta PQR)}}{{ar(\Delta ABC)}} =...

### If in ∆ABC and ∆PQR, we have: , then (a) ∆PQR ~ ∆CAB (b) ∆PQR ~ ∆ABC (c) ∆CBA ~ ∆PQR (d) ∆BCA ~ ∆PQR

Correct Answer: (a) ∆PQR ~ ∆CAB Explanation: In ∆ABC and ∆PQR, $\begin{array}{l} \frac{{AB}}{{QR}} = \frac{{BC}}{{PR}} = \frac{{CA}}{{PQ}}\\ \end{array}$ ∆ABC ~ ∆QRP

### In ∆ABC and ∆DEF, it is given that ∠B = ∠E, ∠F = ∠C and AB = 3DE, then the two triangles are (a) congruent but not similar (b) similar but not congruent (c) neither congruent nor similar (d) similar as well as congruent

Correct Answer: (b) similar but not congruent Explanation: In ∆ABC and ∆DEF, ∠???? = ∠???? ∠???? = ∠???? Applying AA similarity theorem, ∆ABC - ∆DEF. AB = 3DE AB ≠ DE ∆ABC and ∆DEF are similar but...

### If ∆ABC~∆EDF and ∆ABC is not similar to ∆DEF, then which of the following is not true? (a) BC.EF = AC.FD (b) AB.EF = AC.DE (c) BC.DE = AB.EF (d) BC.DE = AB.FD

Correct Answer: (c) BC. DE = AB. EF Explanation: ∆ABC ~ ∆EDF $\begin{array}{l} \frac{{AB}}{{DE}} = \frac{{AC}}{{EF}} = \frac{{BC}}{{DF}}\\ BC.DE \ne AB.EF \end{array}$

### In ∆ABC and ∆DEF, it is given that , then (a) ∠B = ∠E (b) ∠A = ∠D (c) ∠B = ∠D (d) ∠A = ∠F

Correct Answer: (c)∠???? = ∠D Explanation: ∆ ABC − EDF The corresponding angles, ∠???? ???????????? ∠???? ???????????????? ???????? ????????????????????. ∠???? = ∠D

### In ∆ABC, AB = 6√3 , AC = 12 cm and BC = 6cm. Then ∠B is

Answer: Given, ???????? = 6√3???????? ????????2 = 108 ????????2 AC = 12 cm ????????2 = 144 ????????2 BC = 6 cm ????????2 = 36 ???????? ∴ ????????2 = ????????2 + ????????2 The square of the longest...

### In ∆ABC, DE ║ BC such that . If AC = 5.6cm, then AE = ? (a) 4.2cm (b) 3.1cm (c) 2.8cm (d) 2.1cm

Correct Answer: (d) 2.1 cm Explanation:       Given, DE || BC. Applying Thales’ theorem, $\begin{array}{l} \frac{{AD}}{{DB}} = \frac{{AE}}{{EC}}\\ \end{array}$ AE be x cm. EC = (5.6 –...

### The line segments joining the midpoints of the adjacent sides of a quadrilateral form (a) parallelogram (b) trapezium (c) rectangle (d) square

Correct Answer: (a) parallelogram Explanation: The line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

### If the diagonals of a quadrilateral divide each other proportionally, then it is a (a) parallelogram (b) trapezium (c) rectangle (d) square

Correct Answer: (b) trapezium Explanation: Diagonals of a trapezium divide each other proportionally.

### The lengths of the diagonals of a rhombus are 24cm and 10cm. The length of each side of the rhombus is (a) 12cm (b) 13cm (c) 14cm (d) 17cm

Correct Answer: (b) 13 cm Explanation:       Let ABCD be the rhombus with diagonals AC and BD intersecting each other at O. AC = 24 cm BD = 10 cm Diagonals of a rhombus bisect each...

### In a rhombus of side 10cm, one of the diagonals is 12cm long. The length of the second diagonal is (a) 20cm (b) 18cm (c) 16cm (d) 22cm

Correct Answer: (c) 16 cm Explanation:         Let ABCD be the rhombus with diagonals AC and BD intersecting each other at O. Also, diagonals of a rhombus bisect each other at...

### ∆ABC is an isosceles triangle with AB = AC = 13cm and the length of altitude from A on BC is 5cm. Then, BC = ? (a) 12cm (b) 16cm (c) 18cm (d) 24cm

Correct Answer: (d) 24 cm Explanation:     In triangle ABC, Let the altitude from A on BC meets BC at D. AD = 5 cm AB = 13 cm D is the midpoint of BC Applying Pythagoras theorem in...

### The height of an equilateral triangle having each side 12cm, is (a) 6√2 cm (b) 6√3m (c) 3√6m (d) 6√6m

Correct Answer: (b) 6√3????m Explanation:       Let ABC be the equilateral triangle with AD as its altitude from A. In right-angled triangle ABD, ????????2 = ????????2 + ????????2...

### The hypotenuse of a right triangle is 25cm. The other two sides are such that one is 5cm longer than the other. The lengths of these sides are (a) 10cm, 15cm (b) 15cm, 20cm (c) 12cm, 17cm (d) 13cm, 18cm

Correct Answer: (b) 15 cm, 20 cm Explanation: Given, Length of hypotenuse = 25 cm Let the other two sides be x cm and (x−5) cm. Applying Pythagoras theorem, 252 = ????2 + (???? − 5 ) 2 625 = ????2 +...

### In the given figure, O is the point inside a ∆MNP such that ∆MOP = .OM = 16 cm and OP = 12 cm if MN = 21cm and ∆NMP = then NP=?

Answer:     In right triangle MOP, By using Pythagoras theorem, ????????2 = ????????2 + ????????2 => 122 + 162 => 144 + 256 => 400 MO = 20 cm In right triangle MPN, By using...

### A ladder 25m long just reaches the top of a building 24m high from the ground. What is the distance of the foot of the ladder from the building? (a) 7m (b) 14m (c) 21m (d) 24.5m

Correct Answer: (a) 7 m Explanation:       Let the ladder BC reaches the building at C. Let the height of building where the ladder reaches be AC. BC = 25 m AC = 24 m In right-angled...

### The shadow of a 5m long stick is 2m long. At the same time the length of the shadow of a 12.5m high tree (in m) is (a) 3.0 (b) 3.5 (c) 4.5 (d) 5.0

Correct Answer: (d) 5.0 Explanation:         Suppose DE is a 5 m long stick and BC is a 12.5 m high tree. Suppose DA and BA are the shadows of DE and BC. In ∆ABC and ∆ADE...

### A vertical pole 6m long casts a shadow of length 3.6m on the ground. What is the height of a tower which casts a shadow of length 18m at the same time? (a) 10.8m (b) 28.8m (c) 32.4m (d) 30m

Correct Answer: (d) 30m Explanation:         Let AB and AC be the vertical pole and its shadow, AB = 6 m AC = 3.6 m Let DE and DF be the tower and its shadow. DF = 18 m DE =? In...

### A vertical stick 1.8m long casts a shadow 45cm long on the ground. At the same time, what is the length of the shadow of a pole 6m high? (a) 2.4m (b) 1.35m (c) 1.5m (d) 13.5m

Correct Answer: (c) 1.5m Explanation:       Let AB and AC be the vertical stick and its shadow, AB = 1.8 m AC = 45 cm => 0.45 m Let DE and DF be the pole and its shadow, DE = 6 m...

### Two poles of height 13m and 7m respectively stand vertically on a plane ground at a distance of 8m from each other. The distance between their tops is (a) 9m (b) 10m (c) 11m (d) 12m

Correct Answer: (b) 10 m Explanation:         Let AB and DE be the two poles. AB = 13 m DE = 7 m Distance between their bottoms = BE = 8 m Draw a perpendicular DC to AB from D,...

### A man goes 24m due west and then 10m due both. How far is he from the starting point? (a) 34m (b) 17m (c) 26m (d) 28m

Correct Answer: (c) 26 m Explanation:         The man starts from point A and goes 24 m due west to point B. From here, he goes 10 m due north and stops at C. In right triangle...

### For the following statement state whether true(T) or false (F). The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals

Answer: ABCD is a rhombus having AC and BD its diagonals. The diagonals of a rhombus perpendicular bisect each other. AOC is a right-angled triangle. In right triangle AOC, By using Pythagoras...

### For each of the following statements state whether true(T) or false (F) (i) the ratio of the perimeter of two similar triangles is the same as the ratio of their corresponding medians. (ii) if O is any point inside a rectangle ABCD then

Answers: (i) True       Given, ∆ABC ~ ∆DEF ∠???????????? = ∠???????????? ∠???? = ∠???? (∠???????????? ~ ∆????????????) By AA criterion, ∆ABP and ∆DEQ $\frac{A B}{D E}=\frac{A P}{D Q}$...

### For each of the following statements state whether true(T) or false (F) (i) In a ABC , AB = 6 cm, A  and AC = 8 cm and in a DEF , DF = 9 cm  D = and DE= 12 cm, then  ABC ~  DEF. (ii) the polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus.

Answers: (i) False In ∆ABC, AB = 6 cm ∠???? = 450 ???????? = 8 ???????? I???? ∆????????????, ???????? = 9 ???????? ∠???? = 450 ???????? = 12 ???????? ∆???????????? ~ ∆????????????   (ii) False...

### For each of the following statements state whether true(T) or false (F) (i) if two triangles are similar then their corresponding angles are equal and their corresponding sides are equal (ii) The length of the line segment joining the midpoints of any two sides of a triangles is equal to half the length of the third side.

Answers: (i) False If two triangles are similar, their corresponding angles are equal and their corresponding sides are proportional. (ii) True       ABC is a triangle with M, N DE is...

### For each of the following statements state whether true(T) or false (F) (i) Two circles with different radii are similar. (ii) any two rectangles are similar

Answers: (i) False Two rectangles are similar if their corresponding sides are proportional. (ii) True Two circles of any radii are similar to each other.

### Find the length of each side of a rhombus are 40 cm and 42 cm. find the length of each side of the rhombus.

Answer: ABCD is a rhombus. The diagonals of a rhombus perpendicularly bisect each other. ∠???????????? = 900 ???????? = 20 ???????? ???????? = 21 ???????? In right...

### In the given figure, ∠ AMN = ∠ MBC = . If p, q and r are the lengths of AM, MB and BC respectively then express the length of MN of terms of P, q and r.

Answer: In ∆AMN and ∆ABC, ∠???????????? = ∠???????????? =$76^{\circ}$ ∠???? = ∠???? (????????????????????????) By AA similarity criterion, ∆AMN ~ ∆ABC If two triangles...

### Find the length of each side of a rhombus whose diagonals are 24cm and 10cm long.

Answer: ABCD is a rhombus. The diagonals of a rhombus perpendicularly bisect each other. ∠???????????? = 900 ???????? = 12 ???????? ???????? = 5 ???????? In right triangle AOB,...

### Mark (√) against the correct answer in the following: The general solution of the DE is A. B. C. D. None of these

Solution: $\frac{d y}{d x}+y \operatorname{Cot} x=2 \operatorname{Cos} x$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor is $e^{\int \cot x d x}=\operatorname{Sin} x$ General solution is...

### Mark (√) against the correct answer in the following: The general solution of the A. B. C. D. None of these

Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \tan \mathrm{x}=\mathrm{Secx}$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor $e^{\int \tan x d x}=\operatorname{Sec} x$ General...

### In a trapezium ABCD, it is given that AB║CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(∆AOB) = 84cm2 . Find ar(∆COD).

Answer: In ∆ AOB and COD ∠???????????? = ∠???????????? (???????????????????????????????? ???????????????????????? ???????? ???????? ∥ ????????) ∠???????????? = ∠????????????...

### Mark (√) against the correct answer in the following: The general solution of the DE is A. B. C. D. None of these

Solution: $\begin{array}{r} 2 \mathrm{xydy}+\left(\mathrm{x}^{2} \quad \mathrm{y}^{2}\right) \mathrm{d} \mathrm{x}=0 \\ \frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y} \end{array}$ Let...

### Find the length of the altitude of an equilateral triangle of side 2a cm.

Answer: The altitude of an equilateral triangle bisects the side on which it stands and forms right angled triangles with the remaining sides. ABC is an equilateral...

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

Answer: Let AB be A ladder and B is the window at 8 m above the ground C. In right triangle ABC By using Pythagoras theorem, ????????2 = ????????2 + ????????2 102 = 82 +...

### Mark (√) against the correct answer in the following: The general solution od the DE is A. B. C. D. None of these

Solution: $\mathrm{x} \frac{d y}{d x}=y+x \tan \frac{y}{x}$ Dividing both sides by $x$, we get, $\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}$ Let $\mathrm{y}=\mathrm{vx}$ Differentiating both...

### In the given figure, DE║BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.

Answer: In ∆ADE and ∆ABC, ∠???????????? = ∠???????????? (???????????????????????????????????????????????????? ???????????????????????? ???????? ???????? ∥ ????????)...

### Two triangles ABC and PQR are such that AB = 3 cm, AC = 6cm, ∠???? = , PR = 9cm ∠???? = and PQ = 4.5 cm. Show that ∆ ABC ~ ∆ PQR and state that similarity criterion.

Answer: In ∆ABC and ∆PQR ∠???? = ∠???? = 700 $\frac{A B}{P Q}=\frac{A C}{P R}$ $\frac{3}{4.5}=\frac{6}{9}$ $\frac{1}{1.5}=\frac{1}{1.5}$ By SAS similarity criterion, ∆ABC ~...

### State Pythagoras theorem

Pythagoras theorem: The square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, the hypotenuse is the longest side and it’s always opposite the right angle.

### State the SAS-similarity criterion

SAS-similarity criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar.

### Mark (√) against the correct answer in the following: The general solution of the DE is A. B. C. D. None of these

Solution: $\log \left(\frac{d y}{d x}\right)=(a x+b y)$ $\begin{array}{l} \frac{d y}{d x}=e^{a x+b y} \\ \frac{d y}{e^{b y}}=e^{a x} d x \end{array}$ On integrating on both sides we get...

### State the SSS-similarity criterion for similarity of triangles

SSS-similarity criterion for similarity of triangles: If the corresponding sides of two triangles are proportional then their corresponding angles are equal, and hence the two triangles are...

### Mark (√) against the correct answer in the following: The general solution of the is A. B. C. D. None of these

Solution: $\begin{array}{r} x \sqrt{1+y^{2}} \mathrm{dx}+\mathrm{y} \sqrt{1+x^{2}} \mathrm{dy}=0 \\ \frac{y d y}{\sqrt{1+y^{2}}}=\frac{-x d x}{\sqrt{1+x^{2}}} \end{array}$ Let $1+y^{2}=t$ and...

### State the AA-similarity criterion

AA-similarity criterion: If two angles are correspondingly equal to the two angles of another triangle, then the two triangles are similar.

### State the AAA-similarity criterion

AAA-similarity criterion: If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and hence the triangles are similar.

### State the midpoint theorem

Midpoint theorem: The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is equal to one half of the third side.

### State the basic proportionality theorem.

Basic proportionality theorem: If a line is draw parallel to one side of a triangle intersect the other two sides, then it divides the other two sides in the same ratio.

### Mark (√) against the correct answer in the following: the solution of the DE is A. B. C. D. none of these

Solution: $\begin{array}{l} x \operatorname{Cosydy}=\left(x e^{x} \log x+e^{x}\right) d x \\ \operatorname{Cosydy}=\frac{x \operatorname{exlogx}+e x}{x} d x \end{array}$ On integrating on both sides...

### State the two properties which are necessary for given two triangles to be similar.

Answer: The two triangles are similar if and only if The corresponding sides are in proportion. The corresponding angles are equal.

### Mark (√) against the correct answer in the following: The solution of the is A. B. C. D. none of these

Solution: $\cos x(1+\cos y) d x-\sin y(1+\sin x) d y=0$ Let $1+\cos y=t$ and $1+\sin x=u$ On differentiating both equations, we get $-\sin y d y=d t$ and $\cos x d x=d u$ Substitute this in the...

### Mark (√) against the correct answer in the following: The solution of the is A. B. C. D. None of these

Solution: $x \frac{d y}{d x}=\cot y$ Separating the variables, we get, $\begin{array}{c} \frac{d y}{\cot y}=\frac{d x}{x} \\ \tan y d y=\frac{d x}{x} \end{array}$ Integrating both sides, we get,...

### In the given figure, D is the midpoint of side BC and AE⊥BC. If BC = a, AC = b, AB = c, AD = p and AE = h, prove that (i) (ii)

Answers: (i) In right-angled triangle AEC, Applying Pythagoras theorem, ????????2 = ????????2 + ????C2 $b^{2}=h^{2}+\left(x+\frac{a}{2}\right)^{2}$ …(1) In right – angled...

### In the given figure, CD ⊥ AB Prove that

Answer: Given, ∠???????????? = 90o ???????? ⊥ ????B In ∆ ACB and ∆ CDB ∠???????????? = ∠???????????? = 90o ∠???????????? = ∠???????????? By AA similarity-criterion, ∆ ACB ~ ∆CDB...

### In ∆ABC, D is the midpoint of BC and AE⊥BC. If AC>AB, show that

Answer: In right-angled triangle AED, applying Pythagoras theorem, ????????2 = ????????2 + ????????2 … (????) In right-angled triangle AED, applying Pythagoras theorem,...

### Find the length of each side of a rhombus whose diagonals are 24cm and 10cm long.

Answer: Let ABCD be the rhombus with diagonals meeting at O. AC = 24 cm BD = 10 cm We know that the diagonals of a rhombus bisect each other at angles. Applying Pythagoras...

### Find the length of a diagonal of a rectangle whose adjacent sides are 30cm and 16cm.

Answer: Let ABCD be the rectangle with diagonals AC and BD meeting at O. AB = CD = 30 cm BC = AD = 16 cm Applying Pythagoras theorem in right-angled triangle ABC, we get:...

### Find the height of an equilateral triangle of side 12cm.

Answer: Let ABC be the equilateral triangle with AD as an altitude from A meeting BC at D. D will be the midpoint of BC. Applying Pythagoras theorem in right-angled triangle...

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

Answer: Let AD, BE and CF be the altitudes of ∆ABC meeting BC, AC and AB at D, E and F, D, E and F are the midpoint of BC, AC and AB, In right-angled ∆ABD, AB = 2a BD = a...

### Find the length of altitude AD of an isosceles ∆ABC in which AB = AC = 2a units and BC = a units.

Answer: In isosceles ∆ ABC, AB = AC = 2a units BC = a unit Let AD be the altitude drawn from A that meets BC at D. D is the midpoint of BC. BD = BC = ????2...

### ∆ABC is an isosceles triangle with AB = AC = 13cm. The length of altitude from A on BC is 5cm. Find BC.

Answer: Given, ∆ ABC is an isosceles triangle. AB = AC = 13 cm The altitude from A on BC meets BC at D. D is the midpoint of BC. AD = 5 cm ∆ ????????????...

### In the given figure, O is a point inside a ∆PQR such that ∠PQR such that ∠POR = 90o, OP = 6cm and OR = 8cm. If PQ = 24cm and QR = 26cm, prove that ∆PQR is right-angled.

Answer: Applying Pythagoras theorem in right-angled triangle POR, ????????2 = ????????2 + ????????2 ????????2 = 62 + 82 =>36 + 64 =>100 ???????? = √100 =>10 ???????? In...

### A guy wire attached to a vertical pole of height 18 m is 24m 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?

Answer: Let AB be a guy wire attached to a pole BC of height 18 m. Now, to keep the wire taut let it to be fixed at A. In right triangle ABC By using Pythagoras theorem,...

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

Answer: Let the two poles be DE and AB and the distance between their bases be BE. DE = 9 m AB = 14 m BE = 12 m Draw a line parallel to BE from D, meeting AB at C. DC = 12 m AC...

### A ladder is placed in such a way that its foot is at a distance of 15m from a wall and its top reaches a window 20m above the ground. Find the length of the ladder.

Answer: Let the height of the window from the ground and the distance of the foot of the ladder from the wall be AB and BC, AB = 20 m BC = 15 m Applying Pythagoras theorem in...

### A man goes 80m due east and then 150m due north. How far is he from the starting point?

Answer: Let the man starts from point A and goes 80 m due east to B. From B, he goes 150 m due north to c. In right- angled triangle ABC, $A C^{2}=A B^{2}+B C^{2}$...

### The sides of certain triangles are given below. Determine which of them right triangles are (a – 1) cm, 2√???? cm, (a + 1) cm

Answer: Given, The sum of the two sides must be equal to the square of the third side. The three sides of the triangle - a, b and c. P = (a-1) cm, q = 2 √???? ???????? ???????????? ???? = (???? + 1)...

### The sides of certain triangles are given below. Determine which of them right triangles are. (i) 1.4cm, 4.8cm, 5cm (ii) 1.6cm, 3.8cm, 4cm

Answers: Given, The sum of the two sides must be equal to the square of the third side. The three sides of the triangle - a, b and c. (i) a = 1.4 cm b= 4.8 cm c= 5 cm ????2 + ????2 = (1.4) 2 + (4.8)...

### The sides of certain triangles are given below. Determine which of them right triangles are. (i) 9cm, 16cm, 18cm (ii) 7cm, 24cm, 25cm

Answers: Given, The sum of the two sides must be equal to the square of the third side. The three sides of the triangle - a, b and c. (i) a = 9 cm b = 16 cm c = 18 cm ????2 + ????2 = 92 + 162...

### In ∆ABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ∆ADE and ∆ABC.

Answer: Given, D and E are midpoints of AB and AC. Applying midpoint theorem, DE ‖ BC. By B.P.T., $\frac{A D}{A B}=\frac{A E}{A C}$ ∠???? = ∠???? Applying SAS similarity...

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

Answer: Given, DE || BC. ∠???????????? = ∠???????????? (???????????????????????????????????????????????????? ????????????????????????) ∠???????????? = ∠????????????...

### ∆ABC is right angled at A and AD⊥BC. If BC = 13cm and AC = 5cm, find the ratio of the areas of ∆ABC and ∆ADC.

Answer: In ∆ABC and ∆ADC, ∠???????????? = ∠???????????? = 900 ∠???????????? = ∠???????????? (????????????????????????) By AA similarity, ∆ BAC~ ∆ ADC. The ratio of the areas of...

### In the given figure, DE║BC. If DE = 3cm, BC = 6cm and ar(∆ADE) = 15cm2, find the area of ∆ABC.

Answer: Given, DE || BC ∴ ∠???????????? = ∠???????????? (???????????????????????????????????????????????????? ????????????????????????) ∠???????????? = ∠????????????...

### In the given figure, ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1cm, PB = 3cm, AQ = 1.5cm, QC = 4.5cm, prove that area of ∆APQ is 1/16 of the area of ∆ABC.

Answer: Given, $\frac{A P}{A B}=\frac{1}{1+3}=\frac{1}{4}$ $\frac{A Q}{A C}=\frac{1.5}{1.5+4.5}=\frac{1.5}{6}=\frac{1}{4}$ $\frac{A P}{A B}=\frac{A Q}{A C}$ ∠???? = ∠???? By SAS...

### The areas of two similar triangles are 64cm2 and 100cm2 respectively. If a median of the smaller triangle is 5.6cm, find the corresponding median of the other.

Answer: Let the two triangles be ABC and PQR with medians AM and PN, The ratio of areas of two similar triangles will be equal to the ratio of squares of their corresponding...

### The areas of two similar triangles are 81cm2 and 49cm2 respectively. If the altitude of the first triangle is 6.3cm, find the corresponding altitude of the other.

Answer: Given, Triangles are similar The areas of these triangles will be equal to the ratio of squares of their corresponding sides. The ratio of areas of two similar triangles...

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

Answer: Let the two triangles be ABC and DEF with altitudes AP and DQ, Given, ∆ ABC ~ ∆ DEF. The ratio of areas of two similar triangles is equal to the ratio of squares of...

### ∆ABC ~ ∆DEF and their areas are respectively 100cm2 and 49cm2 . If the altitude of ∆ABC is 5cm, find the corresponding altitude of ∆DEF.

Answer: Given, ∆ABC ~ ∆DEF. The ration of the areas of these triangles will be equal to the ratio of squares of their corresponding sides. The ratio of areas of two similar...

### The areas of two similar triangles are 169cm2 and 121cm2 respectively. If the longest side of the larger triangle is 26cm, find the longest side of the smaller triangle.

Answer: Given, Triangles are similar. The ratio of the areas of these triangles will be equal to the ratio of squares of their corresponding sides. Let the longest side of smaller triangle be X cm....

### ∆ABC~∆PQR and ar(∆ABC) = 4, ar(∆PQR). If BC = 12cm, find QR.

Answer: Given, ???????? ( ∆ ???????????? ) = 4???????? (∆ ???????????? ) $$\frac{\operatorname{ar}(\triangle A B C)}{\operatorname{ar}(\Delta P Q R)}=\frac{4}{1}$$ ∵ ∆ABC...

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

Answer: Given, ∆ ABC ~ ∆ PQR The ratio of the areas of triangles will be equal to the ratio of squares of their corresponding sides. \frac{\operatorname{ar}(\triangle A B...