Chapter 1 Similarity Textbook Solution MATHEMATICS Part -2 Class 10 Maharashtra Board

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Practice Set 1.1 | Q 1 | Page 5
Base of a triangle is 9 and height is 5. Base of another triangle is 10 and height is 6. Find the ratio of areas of these triangles.

Practice Set 1.1 | Q 2 | Page 6
In the given figure, BC ⊥ AB, AD ⊥ AB, BC = 4, AD = 8, then find `(A(∆AB))/(A(∆ADB))

Practice Set 1.1 | Q 3 | Page 6
In adjoining figure, seg PS ⊥ seg RQ, seg QT ⊥ seg PR. If RQ = 6, PS = 6 and PR = 12, then Find QT.

Practice Set 1.1 | Q 4 | Page 6
In the following figure, AP ⊥ BC, AD || BC, then Find A(∆ABC): A(∆BCD).

Practice Set 1.1 | Q 5 | Page 6
In adjoining figure, PQ ⊥ BC, AD⊥ BC then find following ratios.

(i) `(“A”(∆”PQB”))/(“A”(∆”PBC”))`

(ii)`(“A”(∆”PBC”))/(“A”(∆”ABC”))`

(iii) `(“A”(∆”ABC”))/(“A”(∆”ADC”))`

(iv) `(“A”(∆”ADC”))/(“A”(∆”PQC”))`

Practice Set 1.2 | Q 1.1 | Page 13

Given below are some triangles and lengths of line segments.
Identify, ray PM is the bisector of ∠QPR.

Practice Set 1.2 | Q 1.2 | Page 13

Given below are some triangles and lengths of line segments.
Identify ray PM is the bisector of ∠QPR.

Practice Set 1.2 | Q 1.3 | Page 13

Given below are some triangles and lengths of line segments.
Identify ray PM is the bisector of ∠QPR

Practice Set 1.2 | Q 2 | Page 13
In ∆PQR, PM = 15, PQ = 25 PR = 20, NR = 8. State whether line NM is parallel to side RQ. Give reason.

Practice Set 1.2 | Q 3 | Page 14
In ∆MNP, NQ is a bisector of ∠N. If MN = 5, PN = 7 MQ = 2.5 then Find QP.

Practice Set 1.2 | Q 4 | Page 14
Measures of some angles in the figure are given. Prove that `”AP”/”PB” = “AQ”/”QC”`

Practice Set 1.2 | Q 5 | Page 14
In trapezium ABCD, side AB || side PQ || side DC, AP = 15, PD = 12, QC = 14, Find BQ.

Practice Set 1.2 | Q 6 | Page 14
Find QP using given information in the figure.

Practice Set 1.2 | Q 7 | Page 14
In the given figure, if AB || CD || FE then Find x and AE.

Practice Set 1.2 | Q 8 | Page 15
In ∆LMN, ray MT bisects ∠LMN If LM = 6, MN = 10, TN = 8, then Find LT.

Practice Set 1.2 | Q 9 | Page 15
In ∆ABC, seg BD bisects ∠ABC. If AB = x, BC = x + 5, AD = x – 2, DC = x + 2, then find the value of x.

Practice Set 1.2 | Q 10 | Page 15
In the given figure, X is any point in the interior of triangle. Point X is joined to vertices of triangle. Seg PQ || seg DE, seg QR || seg EF. Fill in the blanks to prove that, seg PR || seg DF.

Proof : In Δ XDE, PQ || DE …….. ___________

`therefore “XP”/([ ]) = ([ ])/”QE”` ….. (I) (Basic proportionality theorem)

In Δ XEE, QR || EF …….. _________

`therefore ([ ])/([ ]) = ([ ])/([ ])` …….(II) _________________

`therefore ([ ])/([ ]) = ([ ])/([ ])` ……. from (I) and (II)

∴ seg PR || seg DE ……….. (converse of basic proportionality theorem)

Practice Set 1.2 | Q 11 | Page 15
In ∆ABC, ray BD bisects ∠ABC and ray CE bisects ∠ACB. If seg AB ≅ seg AC then prove that ED || BC.

Practice Set 1.3 | Q 1 | Page 21
In the given figure, ∠ABC = 75°, ∠EDC = 75° state which two triangles are similar and by which test? Also write the similarity of these two triangles by a proper one to one correspondence.

Practice Set 1.3 | Q 2 | Page 21
Are the triangles in the given figure similar? If yes, by which test ?

Practice Set 1.3 | Q 3 | Page 21
As shown in figure, two poles of height 8 m and 4 m are perpendicular to the ground. If the length of shadow of smaller pole due to sunlight is 6 m then how long will be the shadow of the bigger pole at the same time?

Practice Set 1.3 | Q 4 | Page 21
In ∆ABC, AP ⊥ BC, BQ ⊥ AC B- P-C, A-Q – C then prove that, ∆CPA ~ ∆CQB. If AP = 7, BQ = 8, BC = 12 then Find AC.

Practice Set 1.3 | Q 5 | Page 22
In trapezium PQRS, side PQ || side SR, AR = 5AP, AS = 5AQ then prove that, SR = 5PQ

Practice Set 1.3 | Q 6 | Page 22
In trapezium ABCD, side AB || side DC, diagonals AC and BD intersect in point O. If AB = 20, DC = 6, OB = 15 then Find OD.

Practice Set 1.3 | Q 7 | Page 22
◻ABCD is a parallelogram point E is on side BC. Line DE intersects ray AB in point T. Prove that DE × BE = CE × TE.

Practice Set 1.3 | Q 8 | Page 22
In the given figure, seg AC and seg BD intersect each other in point P and `”AP”/”CP” = “BP”/”DP”`. Prove that, ∆ABP ~ ∆CDP.

Practice Set 1.3 | Q 9 | Page 22
In the given figure, in ∆ABC, point D on side BC is such that, ∠BAC = ∠ADC. Prove that, CA2 = CB × CD

Practice Set 1.4 | Q 1 | Page 25
The ratio of corresponding sides of similar triangles is 3 : 5; then Find the ratio of their areas..

Practice Set 1.4 | Q 2 | Page 25
If ∆ABC ~ ∆PQR and AB : PQ = 2 : 3, then fill in the blanks.

\[\frac{A\left( ∆ ABC \right)}{A\left( ∆ PQR \right)} = \frac{{AB}^2}{......} = \frac{2^2}{3^2} = \frac{......}{.......}\]

Practice Set 1.4 | Q 3 | Page 25
If ∆ABC ~ ∆PQR, A (∆ABC) = 80, A (∆PQR) = 125, then fill in the blanks.

\[\frac{A\left( ∆ ABC \right)}{A\left( ∆ . . . . \right)} = \frac{80}{125} \therefore \frac{AB}{PQ} = \frac{......}{......}\]

Practice Set 1.4 | Q 4 | Page 25
∆LMN ~ ∆PQR, 9 × A (∆PQR ) = 16 × A (∆LMN). If QR = 20 then Find MN.

Practice Set 1.4 | Q 5 | Page 25
Areas of two similar triangles are 225 sq.cm. 81 sq.cm. If a side of the smaller triangle is 12 cm, then Find corresponding side of the bigger triangle.

Practice Set 1.4 | Q 6 | Page 25
∆ABC and ∆DEF are equilateral triangles. If A(∆ABC) : A(∆DEF) = 1 : 2 and AB = 4, find DE.

Practice Set 1.4 | Q 7 | Page 25
In the given figure 1.66, seg PQ || seg DE, A(∆PQF) = 20 units, PF = 2 DP, then Find A(◻DPQE) by completing the following activity.

Problem Set 1 | Q 1.1 | Page 26
Select the appropriate alternative.
In ∆ABC and ∆PQR, in a one to one correspondence

\[\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}\]

∆PQR ~ ∆ABC

∆PQR ~ ∆CAB

∆CBA ~ ∆PQR

∆BCA ~ ∆PQR

Problem Set 1 | Q 1.2 | Page 26
If in ∆DEF and ∆PQR, ∠D ≅ ∠Q, ∠R ≅ ∠E then which of the following statements is false?

\[\frac{EF}{PR} = \frac{DF}{PQ}\] \[\frac{DE}{PQ} = \frac{EF}{RP}\] \[\frac{DE}{QR} = \frac{DF}{PQ}\] \[\frac{EF}{RP} = \frac{DE}{QR}\]

Problem Set 1 | Q 1.3 | Page 26
In ∆ABC and ∆DEF ∠B = ∠E, ∠F = ∠C and AB = 3DE then which of the statements regarding the two triangles is true ?

The triangles are not congruent and not similar

The triangles are similar but not congruent.

The triangles are congruent and similar.

None of the statements above is true.

Problem Set 1 | Q 1.4 | Page 26
∆ABC and ∆DEF are equilateral triangles, A(∆ABC): A(∆DEF) = 1: 2. If AB = 4 then what is length of DE?
2√2
4
8
4√2

Problem Set 1 | Q 1.5 | Page 26
In the given figure, seg XY || seg BC, then which of the following statements is true?

`”AB”/”AC” = “AX”/”AY”`

`”AX”/”XB” =”AY”/”AC”`

`”AX”/”YC” = “AY”/”XB”`

`”AB”/”YC” = “AC”/”XB”`

Problem Set 1 | Q 2.1 | Page 27
In ∆ABC, B – D – C and BD = 7, BC = 20 then Find following ratio.

`”A(∆ ABD)”/”A(∆ ADC)”`

Problem Set 1 | Q 2.2 | Page 27
In ∆ABC, B – D – C and BD = 7, BC = 20 then Find following ratio.

\[\frac{A\left( ∆ ABD \right)}{A\left( ∆ ABC \right)}\]

Problem Set 1 | Q 2.3 | Page 27
In ∆ABC, B – D – C and BD = 7, BC = 20 then Find following ratio.

\[\frac{A\left( ∆ ADC \right)}{A\left( ∆ ABC \right)}\]

Problem Set 1 | Q 3 | Page 27
Ratio of areas of two triangles with equal heights is 2 : 3. If base of the smaller triangle is 6 cm then what is the corresponding base of the bigger triangle ?

Problem Set 1 | Q 4 | Page 27
In the given figure, ∠ABC = ∠DCB = 90° AB = 6, DC = 8 then \[\frac{A \left( ∆ ABC \right)}{A \left( ∆ DCB \right)} = ?\]

Problem Set 1 | Q 5 | Page 27
In the given figure, PM = 10 cm A(∆PQS) = 100 sq.cm A(∆QRS) = 110 sq.cm then Find NR.

Problem Set 1 | Q 6 | Page 27
∆MNT ~ ∆QRS. Length of altitude drawn from point T is 5 and length of altitude drawn from point S is 9. Find the ratio (A(ΔMNT))/(A(ΔQRS)).

Problem Set 1 | Q 7 | Page 28
In the given figure, A – D- C and B – E – C seg DE || side AB If AD = 5, DC = 3, BC = 6.4 then Find BE.

Problem Set 1 | Q 8 | Page 28
In the given figure, seg PA, seg QB, seg RC, and seg SD are perpendicular to line AD. AB = 60, BC = 70, CD = 80, PS = 280 then Find PQ, QR, and RS.

Problem Set 1 | Q 9 | Page 28
In ∆PQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY || QR.

Problem Set 1 | Q 10 | Page 29
In the given fig, bisectors of ∠B and ∠C of ∆ABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find \[\frac{AX}{XY}\]

Problem Set 1 | Q 11 | Page 29
In ▢ABCD, seg AD || seg BC. Diagonal AC and diagonal BD intersect each other in point P. Then show that \[\frac{AP}{PD} = \frac{PC}{BP}\]

Problem Set 1 | Q 12 | Page 29
In the given fig, XY || seg AC. If 2AX = 3BX and XY = 9. Complete the activity to Find the value of AC.

Activity : 2AX = 3BX

∴ `”AX”/”BX” = square/square`

`”AX +BX”/”BX” = (square + square)/square` …(by componendo)

`”AB”/”BX” = square/square` …(I)

ΔBCA ~ ΔBYX … `square` test of similarity,

∴ `”BA”/”BX” = “AC”/”XY”` …(corresponding sides of similar triangles)

∴ `square/square = “AC”/9`

∴ AC = `square` …[From(I)]

Problem Set 1 | Q 13 | Page 29
In the given figure, the vertices of square DEFG are on the sides of ∆ABC. ∠A = 90°. Then prove that DE2 = BD × EC. (Hint: Show that ∆GBD is similar to ∆CFE. Use GD = FE = DE.)