Class 10-Maths ch-13 GeometryContact Number: 9667591930 / 8527521718

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1.

In the triangle PQR, AB is parallel to QR. The ratio of the areas of two similar triangles PAB and PQR id 1:2. Then PQ:AQ = __________________.

(1) $\sqrt{2}:1$

(2) $1:\sqrt{2}-1$

(3) $1:\left(\sqrt{2}+1\right)$

(4) $\sqrt{2}:\sqrt{2}-1$

2.

In the figure given below, equilateral triangle ECB surmounts square ABCD. Find the angle BED represented by x.

(1) $15\xb0$

(2) $30\xb0$

(3) $45\xb0$

(4) $60\xb0$

3.

In two triangles ABC and DEF, $\angle A=\angle D$. The sum of the angles A and B is equal to the sum of the angles D and E. If BC = 6 cm and EF = 8 cm, find the ratio of the area of the triangles, ABC and DEF.

(1) 3 : 4

(2) 4 : 3

(3) 9 : 16

(4) 16 : 9

4.

In the following figure, PQ is parallel to BC and PQ : BC = 1 : 3. If the area of the triangle ABC is 144 $c{m}^{2}$, then what is the area of the triangle APQ?

(1) 48 ${\mathrm{cm}}^{2}$

(2) 36 ${\mathrm{cm}}^{2}$

(3) 16 ${\mathrm{cm}}^{2}$

(4) 9 ${\mathrm{cm}}^{2}$

5.

In triangle ABC, sides AB and AC are extended to D and E, respectively, such that AB = BD and AC = CE. Find DE, if BC = 6 cm.

(1) 3 cm

(2) 6 cm

(3) 9 cm

(4) 12 cm

6.

A man travels on a bicycle, 10 km east from the starting point A to reach point B, then he cycles 15 km south to reach point C. Find the shortest distance between A and C.

(1) 25 km

(2) 5 km

(3) $\sqrt[25]{13}$ km

(4) $\sqrt[5]{13}$ km

7.

$Inthefollowingfigure,Oisthecentreofthecircle.If\angle BAC=60\xb0,then\angle OBC=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

(1) $120\xb0$

(2) $30\xb0$

(3) $40\xb0$

(4) $60\xb0$

8.

$Inthefollowingfigure(nottoscale),AB=CDand\overline{AB}and\overline{CD}areproducedtomeetatthepointP.\phantom{\rule{0ex}{0ex}}If\angle BAC=70\xb0,thenfind\angle P.$

(1) $30\xb0$

(2) $40\xb0$

(3) $45\xb0$

(4) $50\xb0$

9.

PT and PS are the tangents to the circle with centre O. If $\angle TPS=65\xb0,then\angle OTS=\_\_\_\_\_\_\_\_\_\_\_\_\_$

(1) $32\xb0$

(2) $45\xb0$

(3) $57\frac{1}{2}\xb0$

(4) $32\frac{1}{2}\xb0$

10.

In the following figure X, Y and Z are the points at which the incircle touches the sides of the triangle. If PX = 4 cm, QZ = 7 cm and YR = 9 cm, then the perimeter of triangle PQR is

(1) 20 cm

(2) 46 cm

(3) 40 cm

(4) 80 cm

11.

The locus of the point P which is at a constant distance of 2 units from the origin and which lies in the first or the second quadrants is_______________

(1) $y=-\sqrt{4-{x}^{2}}$

(2) $y=\sqrt{4-{x}^{2}}$

(3) $x=\sqrt{4-{y}^{2}}$

(4) $x=-\sqrt{4-{y}^{2}}$

12.

$IfPABisatriangleinwhich\angle B=90\xb0andA(1,1)andB(0,1),thenthelocusofPis\_\_\_\_\_\_\_\_\_\_\_\_\_$

(1) y = 0

(2) xy = 0

(3) x = y

(4) x = 0

13.

In the following figure, if the angle between two chords AB and AC is $65\xb0$, then the angle between two tangents which are drawn at B and C is ________________

(1) $50\xb0$

(2) $30\xb0$

(3) $60\xb0$

(4) $40\xb0$

14.

In the following figure, O is the centre of the circle and $\angle AMB=120\xb0$, find the angle between the two tangents AP and BP.

(1) $30\xb0$

(2) $45\xb0$

(3) $70\xb0$

(4) $60\xb0$

15.

If ABCD is a square inscribed in a circle and PA is a tangent, then the angle between the lines P'A and P'B is_____________

(1) $30\xb0$

(2) $20\xb0$

(3) $40\xb0$

(4) $45\xb0$

16.

In the following figure, if l and m are two tangents and AB is a chord making an angle of $60\xb0$ with the tangent l, then the angle between l and m is____________

(1) $45\xb0$

(2) $30\xb0$

(3) $60\xb0$

(4) $90\xb0$

17.

Find the length of a transverse common tangent of the two circles whose radii are 3.5 cm, 4.5 cm and the distance between their centres is 10 cm.

(1) 36 cm

(2) 6 cm

(3) 64 cm

(4) 8 cm

18.

If ABCD is a trapezium, AC and BD are the diagonals interesting each other at point O. Then AC:BD = __________

(1) AB : CD

(2) AB + AD : DC + BC

(3) $A{O}^{2}:O{B}^{2}$

(4) AO - OC : OB - OD

19.

In the following figure (not to scale), $\overline{PA}and\overline{PB}$ are equal chords and ABCD is a cyclic quadrilateral. If $\angle DCE=80\xb0,\angle DAP=30\xb0,thenfind\angle APB.$

(1) $40\xb0$

(2) $80\xb0$

(3) $90\xb0$

(4) $160\xb0$

20.

In trapezium KLMN, KL and MN are parallel sides. A line is drawn, from the point A on KN, parallel to MN meeting LM at B. KN : LM is equal to ____________________

(1) KL : NM

(2) (KL + KA) : (NM + BM)

(3) (KA - AN) : (LB - BM)

(4) $K{L}^{2}:M{N}^{2}$

21.

In the following figure, ABCD is a square and AED is an equilateral triangle. Find the value of a.

(1) $30\xb0$

(2) $45\xb0$

(3) $60\xb0$

(4) $75\xb0$

22.

A circle with centre O is inscribed in a quardrilateral ABCD as shown in the figure. Which of the following statements is / are true?

(A) $\angle AOD+\angle BOC=180\xb0$

(B) $\angle AOBand\angle CODarecomplementary$

(C) $OA,OB,OCandODaretheanglebisectorsof\angle A,\angle B,\angle Cand\angle Drespectively.$

(1) Both (A) and (B)

(2) Both (B) and (C)

(3) Both (A) and (C)

(4) all the three

23.

In the following figure, O is the centre of the circle and if $\angle OAB=30\xb0$, then the acute angle between AB and the tangent PQ at B is _______________.

(1) $30\xb0$

(2) $60\xb0$

(3) $45\xb0$

(4) $90\xb0$

24.

In the following figure, AB = OB and CT is the tangent to the circle at O. If $\angle COA=125\xb0,then\angle OABis\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

(1) $55\xb0$

(2) $27\frac{1}{2}\xb0$

(3) $82\frac{1}{2}\xb0$

(4) $45\xb0$

25.

AR and BS are the tangents to the circle, with centre O, touching at P and Q respectively and PQ is the chord. If $\angle OQP=25\xb0,then\angle RPQ=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

(1) $100\xb0$

(2) $115\xb0$

(3) $150\xb0$

(4) $90\xb0$

26.

$Inthediagram,if\angle BCP\text{'}=\angle ABQ=60\xb0,thenthetriangleABCis\_\_\_\_\_\_\_\_\_\_\_\phantom{\rule{0ex}{0ex}}$

(1) scalene

(2) equilateral

(3) right angled

(4) acute angled

27.

In the following figure, AQ is a tangent to the circle at A. If $\angle ACB=60\xb0,then\angle BAQ=\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

(1) $30\xb0$

(2) $60\xb0$

(3) $120\xb0$

(4) $45\xb0$

28.

In the following figure, two circles X and Y with centres A and B respectively interesect at C and D. The radii AC and AD of circle X are tangents to the circle Y. Radii BC and BD of circle Y are tangents to the circle X. Find $\angle AEC.$

(1) $45\xb0$

(2) $60\xb0$

(3) $90\xb0$

(4) Cannot be determined

29.

The tangent AB touches a circle, with centre O, at the point P. If the radius of the circle is 5 cm, OB = 10 cm and OB = AB, then find AP.

(1) $5\sqrt{5}cm$

(2) $10\sqrt{5}cm$

(3) $\left(10-5\sqrt{3}\right)cm$

(4) $\left(10-\frac{5}{\sqrt{3}}\right)cm$

30.

In the following figure, O is the centre of the circle and D, E and F are mid-points of AB, BO and OA respectively. If $\angle DEF=30\xb0,thenfind\angle ACB.$

(1) $30\xb0$

(2) $60\xb0$

(3) $90\xb0$

(4) $120\xb0$

31.

In the figure given below, ABC is an equilateral triangle and PQRS is a square of side 6 cm. By how many $c{m}^{2}$ is the area of the triangle more than that of the square ?

(1) $\frac{21}{\sqrt{3}}$

(2) 21

(3) $21\sqrt{3}$

(4) 63

32.

In the following figure (not to scale) ABCD is a rectangle, BC = 24 cm, DP = 10 cm and CD = 15 cm. Then, AQ and CQ respectively are ______________.

(1) 39 cm, 13 cm

(2) 13 cm, 12 cm

(3) 25 cm, 13 cm

(4) 39 cm, 12 cm

33.

At a particular time, the shadow cast by a tower is 6 m long. If the distance from top of the tower to the end of the shadow is 10 m long, determine the height of the tower.

(1) 4 m

(2) 8 m

(3) 16 m

(4) 12 m

34.

$Inthefollowingfigure,\angle ABC=90\xb0,AD=15andDC=20.IfBDisthebisectorof\angle ABC,\phantom{\rule{0ex}{0ex}}whatistheperimeterofthetriangleABC?$

(1) 74

(2) 84

(3) 91

(4) 105

35.

$InthetriangleABC,\angle ABCor\angle B=90\xb0.AB:BD:DC=3:1:3.IfAC=20cm\phantom{\rule{0ex}{0ex}}thenwhatisthelengthofAD(incm)?$

(1) $5\sqrt{2}$

(2) $6\sqrt{3}$

(3) $4\sqrt{5}$

(4) $4\sqrt{10}$

36.

In the following figure, AD

(1) 1.8 cm

(2) 2.25 cm

(3) 2.2 cm

(4) 1.85 cm

37.

In the given figure, AB is a diameter, O is the centre of the circle and $\angle OCB=50\xb0,thenfind\angle DBC.$

(1) $80\xb0$

(2) $100\xb0$

(3) $120\xb0$

(4) $140\xb0$

38.

In the following figure, O is the centre of the circle and AD is the diameter. If $\angle ACB=135\xb0,thenfind\angle DOB.$

(1) $135\xb0$

(2) $60\xb0$

(3) $90\xb0$

(4) $45\xb0$

39.

In the following figure, O is the centre of the circle, AC is the diameter and if $\angle APB=120\xb0,thenfind\angle BQC.$

(1) $30\xb0$

(2) $150\xb0$

(3) $90\xb0$

(4) $120\xb0$

40.

In the trapezium PQRS, PQ is parallel to RS and the ratio of the areas of the triangle POQ to triangle ROS is 225:900. Then SR = _____________.

(1) 30 PQ

(2) 25 PQ

(3) 2 PQ

(4) PQ

41.

In the following figure, ABCD is a parallelogram, CB is extended to F and the line joining D and F intersect AB at E. Then ____________.

(1) $\frac{AD}{AE}=\frac{BF}{BE}$

(2) $\frac{AD}{AE}=\frac{CF}{CD}$

(3) $\frac{BF}{BE}=\frac{CF}{CD}$

(4) All of these

42.

$PQRisarightangledtriangle,where\angle P=90\xb0.\overline{PD}isperpendicularto\overline{QR}.\phantom{\rule{0ex}{0ex}}PQ:\sqrt{PR}=\_\_\_\_\_\_\_\_$

(1) QD:$\sqrt{DR}$

(2) $\sqrt{QD}:\sqrt{DR}$

(3) $Q{D}^{2}:\sqrt{R{D}^{2}}$

(4) None of these

43.

Two circles intersect at two points P and S. QR is a tangent to the two circles at Q and R. If $\angle QSR=72\xb0,then\angle QPR=\_\_\_\_\_\_\_\_\_\_\_$

(1) $84\xb0$

(2) $96\xb0$

(3) $102\xb0$

(4) $108\xb0$

44.

An equilateral triangle CDE is constructed on a side CD of square ABCD. The measure of $\angle AEB$can be ________

(1) $150\xb0$

(2) $45\xb0$

(3) $30\xb0$

(4) $20\xb0$

45.

In the figure above (not to scale), PQ is a tangent segment to the circle at P. If p, R, S and T are concyclic points, $\angle QPR=40\xb0andPR=RQ,thenfind\angle PTS$

(1) $80\xb0$

(2) $100\xb0$

(3) $60\xb0$

(4) $120\xb0$

46.

Construct the incircle of a given triangle ABC. The following sentences are the steps involved in the above construction. Arrange them in sequencial order from first to last.

(A) Draw perpendicular $\overline{IM}fromIonto\overline{BC.}$

(B) Taking I as centre and IM as the radius, draw a circle.

(C) Draw a DABC with the given measurements.

(D) Draw bisectors of two of the angles, say $\angle Band\angle CtointersectatI.$

(1) DCAB

(2) CDAB

(3) CADB

(4) DACB

47.

Construct a regular pentagon in a circle of radius 4 cm. The following sentences are the steps involved in the following constructions. Arrange them in sequential order from first to last.

(A) With D as centre and AB as radius draw an arc which cuts the circle at the point C.

(B) With A as centre and AB as radius draw an arc which cuts the circle at the point E.

(C) Construct a circle with radius 4 cm.

(D) Join AB.

(E) Draw two radii $\overline{OA}and\overline{OB}suchthat\angle AOB=72\xb0$

(F) Join AE, ED, DC and CB.

(G) With E as centre and AB as radius draw an arc which cuts the circle at the point D.

(1) CEDGBAF

(2) CEBDGAF

(3) CEDBGAF

(4) CEBGADF

48.

In the diagram, O is the centre of the circle and $\angle OPA=30\xb0$. $Find\angle ACBand\angle ADBrespectively.$

(1) $120\xb0,60\xb0$

(2) $60\xb0,120\xb0$

(3) $75\xb0,105\xb0$

(4) $35\xb0,145\xb0$

49.

Side of a square PQRS is 4 cm long. $\overline{PR}isproducedtothepointMsuchthatPR=2RM.FindSM.$

(1) $\sqrt{10}cm$

(2) $\sqrt{5}cm$

(3) $2\sqrt{5}$cm

(4) $2\sqrt{10}cm$

50.

ABC is an equilateral triangle of side 6 cm. If a circle of radius 1 cm is moving inside and along the sides of the triangle, then locus of the centre of the circle is an equilateral triangle of side _______________.

(1) 5 cm

(2) 4 cm

(3) $(6-2\sqrt{3})cm$

(4) $\left(3+\sqrt{3}\right)cm$

51.

In the following figure (not to scale), STM and MQ are tangents to the circle at T and Q respectively. SRQ is a straight line. SR = TR and $\angle TSR=25\xb0.Find\angle QMT.$

(1) $55\xb0$

(2) $60\xb0$

(3) $75\xb0$

(4) $80\xb0$

52.

PQ is the direct common tangent of two circles (S, 9 cm) and (R, 4 cm) which touch each other externally. Find the area of the quadrilateral PQRD (in $c{m}^{2}$).

(1) 72

(2) 65

(3) 78

(4) 69

53.

Diagonal AC of a rectangle ABCD is produced to the point E such that AC:CE = 2:1. AB = 8 cm and BC = 6 cm. Find the length of DE.

(1) $2\sqrt{19}$

(2) 15 cm

(3) $3\sqrt{17}cm$

(4) 13 cm

54.

$In\u2206PQR,PQ=6cm,PR=9cmandMisapointonQRsuchthatitdivides\phantom{\rule{0ex}{0ex}}QRintheratio1:2.PM\perp QR.FindQR.$

(1) $\sqrt{18}cm$

(2) $3\sqrt[]{12}cm$

(3) $3\sqrt[]{15}cm$

(4) $\sqrt{20}cm$

55.

In the following figure (not to scale), ABCD is an isosceles trapezium. $\overline{AB}\parallel \overline{CD},AB=9cmandCD=1cm.AP:PD=BQ:QC=1:2.FindPQ.$

(1) 11 cm

(2) 10.5 cm

(3) 10 cm

(4) 9.5 cm

56.

P, Q and R are on AB, BC and AC of the equilateral triangle ABC respectively. AP:PB=CQ:QB=1:2. G is the centroid of the triangle PQB and R is the mid-point of AC. Find BG:GR.

(1) 1:2

(2) 2:3

(3) 3:4

(4) 4:5

57.

In the given figure (not to scale), two circles ${C}_{1}$and ${C}_{2}$ intersect at S and Q. PQN and RQM are tangents drawn to ${C}_{1}$ and ${C}_{2}$ respectively at Q. MAB and ABN are the chords of the circles ${C}_{1}$ and ${C}_{2}$. If $\angle NQR=85\xb0$, then find $\angle AQB$.

(1) $5\xb0$

(2) $10\xb0$

(3) $15\xb0$

(4) Cannot be determined

58.

Two sides of a triangle are 5 cm and 12 cm long. The measure of third side is an integer in cm. If the triangle is an obtuse triangle, then how many such triangles are possible?

(1) 9

(2) 8

(3) 7

(4) 6

59.

$Ina\u2206PQR,MliesonPRandbetweenPandRsuchthatQR=QM=Pm.\phantom{\rule{0ex}{0ex}}If\angle MQR=40\xb0,thenfind\angle P.$

(1) $35\xb0$

(2) $25\xb0$

(3) $45\xb0$

(4) $55\xb0$

60.

In the following figure, (not to scale), AB is the common tangent to the circles ${C}_{1}$ and ${C}_{2}$. ${C}_{1}$ and ${C}_{2}$ are touching externally at C. AD and DC are two chords of the circle ${C}_{1}$and BE and CE are two chords of the circle ${C}_{2}$. Find the measure of $\angle ADC+\angle BEC.$

(1) $60\xb0$

(2) $90\xb0$

(3) $120\xb0$

(4) $100\xb0$

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