**Question 13. The perimeters of two similar triangles are 25 cm and 15 cm, respect. If one side of the first triangle is 9 cm, what is the corresponding side of the other triangle?**

**Solution:**

Given: perimeter of two similar triangles are 25 cm, 15 cm and one side 9 cm

To find : the other side.

Let the two triangles be ABC & PQR.

Let one of its side is (AB) = 9 cm and the other side of other triangle be PQ.

âˆ†ABC âˆ¼ âˆ†PQR. [Since the ratio of corresponding sides of similar triangles is same as the ratio of their perimeters.]

AB/PQ = BC/QR = AC/PR = 25/15

9/PQ = 25/15

25 PQ = 15 Ã— 9

PQ = (15Ã—9)/25

PQ =(3Ã— 9 )/5

PQ = 27/5

PQ = 5.4 cm

Hence, the corresponding side of other âˆ† is 5.4 cm.

**Question 14. In Î”ABC and Î”DEF, it is being given that AB = 5 cm, BC = 4 cm, CA = 4.2 cm, DE = 10 cm, EF = 8 cm, and FD = 8.4 cm. If AL âŠ¥ BC, DM âŠ¥ EF, find AL: Dm.**

**Solution:**

Given,

AB = 5 cm, BC = 4 cm, CA = 4.2 cm, DE = 10 cm, EF = 8 cm, and FD = 8.4 cm

AL âŠ¥ BC, DM âŠ¥ EF

To find: AL: DM

Now In âˆ†ABC and âˆ†DEF,

AB/DE=BC/EF=AC/DF=1/2 [both triangles are similar]

[In similar triangle the ratio of corresponding altitude is same as the ratio of the corresponding sides.]

AL: DM = 1: 2

**Question 15. D and E are the points on the sides AB and AC respectively, of a Î”ABC such that AD = 8 cm, DB = 12 cm, AE = 6 cm, and CE = 9 cm. Prove that BC = 5/2 DE.**

**Solution:**

AE/EC=6/9=2/3…….(1)

AD/DB=8/12=2/3……(2)

from (1) and (2) we know that

AD/DB=AE/EC

EDâˆ¥CB

Now Inâ–³AED and â–³ACB,

EDâˆ¥CB

âˆ AED=âˆ ACB

âˆ ADE=âˆ ABC

â–³AEDâˆ¼â–³ACB [by AAA criteria]

AE/AC=ED//CB

AE/EC+AE= ED/CB [AC=AE+EC]

6/6+9= ED/CB

6/15=2/5=ED/CB

CB=5/2ED

Hence proved

**Question 16. D is the midpoint of side BC of a Î”ABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE: EX = 3 : 1**

**Solution:**

Given: In âˆ†ABC, D is the midpoint of BC, E is the midpoint of AD. BE produced meets AC at X.

To prove : BE: EX = 3 : 1

Now In Î” BCX and Î”DCY,

âˆ CBX = Î”CBY (corresponding angles)

âˆ CXB = Î”CYD (corresponding angles)

Î”BCXâˆ¼Î”DCY (AA similarity)

BC/DC = BX/ DY = CX/CY [ corresponding sides of two similar triangles are proportional]

BX/DY = BC/DC

BX/DY = 2DC/DC [As D is the midpoint of BC]

BX/DY = 2/1â€¦â€¦â€¦…(i)

Now In Î”AEX and Î”ADY,

âˆ AEX = Î”ADY (corresponding angles)

âˆ AXE = Î”AYD (corresponding angles)

Î”AEX âˆ¼ Î”ADY (AA similarity)

AE/AD = EX/DY = AX/AY [ corresponding sides of two similar triangles are proportional]

EX/DY = AE/AD

EX/DY = AE/2AE [ D is the midpoint of BC]

EX/DY = Â½â€¦â€¦â€¦â€¦..â€¦(ii)

On Dividing eqn. (i) by eqn. (ii) we get,

BX/EX = 4/1

BX = 4EX

BE + EX = 4EX

BE = 4EX – EX

BE = 3EX

BE /EX = 3/1

BE : EX = 3:1

Hence proved

**Question 17. 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 rectangle contained by AB and BC.**

**Solution:**

Given : ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q.

To prove : BP x DQ = AB x BC

Now In âˆ†ABP and âˆ†QDA,

âˆ B = âˆ D [Opposite angles of parallelogram]

âˆ BAP = âˆ AQD [Alternate interior angles]

âˆ†ABP ~ âˆ†QDA [By AA similarity]

AB/QD = BP/DA [the corresponding sides of similar triangles are proportional]

DA = BC [opposite sides of a parallelogram are equal]

AB/QD = BP/BC

AB Ã— BC = QD Ã— BP

Hence proved

**Question 18. In **Î”**ABC, AL and CM are the perpendiculars from the vertices A and C to BC and AB respect. If Al and CM intersect at O, prove that:**

**(i) Î”OMA âˆ¼ Î”OLC**

**(ii)OA/OC=OM/OL**

**Solution:**

Given,

AL âŠ¥ BC and CM âŠ¥ AB

To prove:

(i) Î”OMA âˆ¼ Î”OLC

(ii)OA/OC=OM/OL

Now In Î” OMA and Î”OLC,

âˆ MOA = âˆ LOC [Vertically opposite angles]

âˆ AMO = âˆ CLO [Each 90Â°]

Î”OMA ~ Î”OLC [By AA similarity]

OA/OC=OM/OL [Corresponding parts of similar Î” are proportional]

Hence proved

**Question 19. ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD**,** and BD respect. Show that PQRS is a rhombus.**

**Solution:**

Given,

ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid points of AB, AC, CD, BD, respectively.

To prove: PQRS is a rhombus

Now,

In Î”ABC, P and Q are the mid points of the sides B and AC respectively

PQ âˆ¥ BC and PQ = 1/2 BC. [By the midpoint theorem]

In Î”ADC, Q and R are the mid points of the sides AC and DC respectively

By the mid-point theorem, we get,

QR âˆ¥ AD QR = 1/2 AD [By the mid-point theorem]

QR = 1/2 BC [AD = BC]

Now In Î”BCD,

RSâˆ¥BC and RS = 1/2 AD [By the mid-point theorem]

RS= 1/2 BC [AD = BC]

From above eqns.

PQ = QR = RS

PQRS is a rhombus.

Hence proved

**Question 20. In an isosceles Î”ABC, the base AB is produced both ways to P and Q such that AP x BQ = AC2. Prove that Î” APC âˆ¼ Î” BCQ**

**Solution:**

Given : Î”ABC is isosceles âˆ† and AP x BQ = ACÂ²

To prove : Î”APCâˆ¼Î”BCQ.

we know that,

Î”ABC is an isosceles triangle AC = BC.

AP x BQ = ACÂ² (given)

AP x BQ = AC x AC

AP x BQ = AC x BC

AP/BC = ABQâ€¦â€¦â€¦.(1).

so, âˆ CAB = âˆ CBA [as AC = BC]

180Â° â€“ âˆ CAP = 180Â° â€“ âˆ CBQ [angles opposite to equal sides are EQUAL]

âˆ CAP = âˆ CBQ â€¦â€¦â€¦..(2)

In âˆ†APC and Î”BCQ

AP/BC = AC/BQ [From equation 1]

âˆ CAP = âˆ CBQ [From equation 2]

Î”APCâˆ¼Î”BCQ [By SAS similarity criterion]

Hence proved

**Question 21. A girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m/sec. If the lamp is 3.6m above the ground, find the length of her shadow after 4 seconds.**

**Solution:**

AD=1.2 m/sec Ã—4 sec =4.8 m =480 cm

Suppose the length of the shadow of the girl be x cm when she at position D. Therefore,

BD = x cm

In âˆ†BDE and âˆ†BAC,

âˆ BDE=âˆ BAC [each90Â°]

âˆ DBE=âˆ ABC [Common angle ]

âˆ†BDEâˆ¼âˆ†BAC [ by AA similarity]

BE/BC=DE/AC=BD/AB [Corresponding sides are proportional]

90/360=x/480+x

1/4= x/480+x

4x=480+x

4xâˆ’x=480

3x=480

x=480/3=160

Hence, the length of her shadow after 4 seconds is 160 cm.

**Question 22. A vertical stick of length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28m long. Find the height of the tower.**

**Solution:**

Let AB be stick and DE be tower.

A 6 m long stick casts a shadow of 4 m

So, AB = 6 m and BC = 4 m

Let DE casts shadow at the same time, So EF = 28 m

Let height of tower DE = x

Now in âˆ†ABC and âˆ†DEF,

âˆ B = âˆ E (each 90Â°)

âˆ C = âˆ F (shadows at the same time)

âˆ†ABC ~ âˆ†DEF [AA criterion]

AB/DE=BC/EF [ Corresponding sides are proportional ]

6/x=4/28

x=42 m

Hence height of tower is 42 m

**Question 23. In the fig. given, **Î”**ABC is a **right-angled** triangle at C and DE âŠ¥ AB. Prove that **Î”**ABC âˆ¼ **Î”**ADE.** **hence find the lengths of AE** **and DE.**

**Solution:**

Given : Î”ACB is right-angled triangle at C = 90Â°.

According to the figure : BC = 12 cm , AD=3 cm, DC = 2 cm.

AC = AD + DC = 3 +2= 5 cm

Now In âˆ†ACB,

ABÂ² = ACÂ² + BCÂ² [by Pythagoras theorem]

ABÂ² = 5Â² + 12Â²

ABÂ² = 25 + 144 = 169

AB= âˆš169 = 13

AB = 13 cm

Now In Î”ABC & Î”ADE,

âˆ BAC = âˆ DAE [common]

âˆ ACB = âˆ AED [each 90Â°]

Î”ABCâˆ¼Î”ADE [by AA similarity criterion]

AB/AD = BC/DE = AC/AE [ corresponding sides of two similar triangles are proportional]

13/3 = 12/ DE = 5/AE

13/3 = 12/DE

13 DE = 12Ã—3

DE = 36/13

13/3 = 5/AE

13 AE = 5Ã—3

AE = 15/13

Hence, the length of DE= 36/13 & AE = 15/13

**Question 24. In the figure, PA, QB and RC are each perpendicular to AC. Prove that 1/X+1/Z=1/Y**

Given:

PA, QB and RC are each perpendicular to AC

To prove : 1/X+1/Z=1/Y

Now In Î”PAC and Î”QBC,

âˆ PCA = âˆ QCB [common angle]

âˆ PAC = âˆ QBC [Corresponding angles]

Î”PAC ~ Î”QBC [by AA criteria]

PA/QB = AC/BC [ Corresponding parts of similar triangles are proportional]

x/y = AB/BC

y/x = BC/AC

Now In Î”RCA and Î”QBA,

âˆ RAC = âˆ QAB [common angle]

âˆ RCA = âˆ QBA [Corresponding angle]

Î”RCA ~Î”QBA [by AA criteria]

RC/QB = AC/AB [ Corresponding parts of similar triangles are proportional]

z/y= AC/AB

y/z= AB/AC

y/z + y/z = BC + AC/ AC = 1 [by adding both eq we get]

y/z + y/z = 1

1/x + 1/z = 1/y [multiplying both sides by y]

Hence proved

**Question 25. In fig. given, we have AB âˆ¥ CD âˆ¥ 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.**

**Solution:**

Given : AB || CD || EF , AB = 6 cm, CD = x cm, BD = 4 cm, and DE = y cm and EF = 10 cm.

Now In âˆ†ECD and âˆ†EAB,

âˆ CED = âˆ AEB [common]

âˆ ECD = âˆ EAB [corresponding angles]

âˆ†ECD ~ âˆ†EAB â€¦â€¦….(1) [By AA similarity]

âˆ´ EC/EA = CD/AB [Corresponding parts of similar triangles are proportional]

EC / EA = x/6 â€¦â€¦â€¦â€¦â€¦(2)

In âˆ†ACD and âˆ†AEF

âˆ CAD = âˆ EAF [common]

âˆ ACD = âˆ AEF [corresponding angles]

âˆ†ACD ~ âˆ†AEF [By AA similarity]

âˆ´ AC/AE = CD/EF [Corresponding parts of similar triangles are proportional]

AC /AE = x/10 â€¦â€¦â€¦â€¦â€¦..(3)

EC/EA+ AC /AE = x/6 + x/10 [by Adding eq 2 & 3]

(EC + AC) /AE =( 5x + 3x)/30

AE / AE = 8x /30

1 = 8x/30

x = 30/8

x = 3.75 cm

âˆ†ECD ~ âˆ†EAB [From eq (i)]

DC/AB = ED /EB [Corresponding parts of similar triangle are proportional]

3.75/6 = y/y+4

6y = 3.75(y+4)

6y = 3.75y + 15

6y – 3.75y = 15

2.25y = 15

y = 15/2.25

y = 6.67 cm

Hence, x = 3.75 cm and y = 6.67 cm.