Class 9 Pearson - GeometryContact Number: 9667591930 / 8527521718

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In the following figure, Δ*ABC *is right-angled at *C*, and *M* is the mid-point of hypotenuse *AB*. If *AC* = 32 cm and *BC *= 60 cm, then find the length of $\overline{CM}$*.*

(1) 32 cm

(2) 30 cm

(3) 17 cm

(4) 34 cm

A cyclic polygon has *n* sides such that each of its interior angle measures 144°. What is the measure of the angle subtended by each of its side at the geometrical centre of the polygon?

(1) 144°

(2) 30°

(3) 36°

(4) 54°

In the following, *PQRS* is a rhombus. *SQ *and *PR *are the diagonals of the rhombus intersecting at *O. *If angle *OPQ* = 35°, then find the value of angle *ORS* + angle *OQP.*

(1) 90°

(2) 180°

(3) 135°

(4) 45°

In the following, *CDEF *is a cyclic quadrilateral. $\overline{CG}$ and $\overline{DH}$* *are the angle bisectors of *∠**C *and ∠*D *respectively. If ∠*E *= 100° and ∠*F *= 110°, then find *∠**CPD.*

(1) 105°

(2) 80°

(3) 150°

(4) 90°

In the following figure, *ABC *is an equilateral triangle. *DE *is parallel to *BC *and equal to half the length of *BC. *If *AD *+ *EC* + *CB* = 24 cm, then what is the perimeter of triangle *ADE*?

(1) 12 cm

(2) 16 cm

(3) 18 cm

(4) Cannot be determined

In Δ*PQR, M* and *N* are points on *PQ* and *PR*, respectively, such that **$\overline{MN}\parallel \overline{QR}$**. If *PM* = *x,* *PR *=* x* + 9*, PQ *=* x + *13 and *PN *=* x* – 2, then find *x*.

(1) 10

(2) 11

(3) 13

(4) 15

In the following figure (not to scale), the chords *AC *and *BD *intersect at *E *and *∠**BAE* = *∠**ECD* + 20°. If *∠**CDE* = 60°, find *∠**ABE.*

(1) 40°

(2) 60°

(3) 80°

(4) None of these

In the following figure, $\overline{DE}$ and $\overline{FG}$* *are equal chords of the circle subtending *∠**DHE *and *∠**FHG *at the point *H *on the circle. If *∠**DHE* = $23{\frac{1}{2}}^{\circ}$, then find *∠**FHG.*

(1) $27{\frac{1}{2}}^{\circ}$

(2) 30°

(3) $23{\frac{1}{2}}^{\circ}$

(4) 60°

The bisectors of two adjacent angles in a parallelogram meet at a point *P *inside the parallelogram. The angle made by these bisectors at a point is _______.

(1) 180°

(2) 90°

(3) 45°

(4) None of these

If *x° *is the measure of an angle which is equal to its complement and *y* is the measure of an angle which is equal to its supplement, then $\frac{{x}^{\circ}}{{y}^{\circ}}$ is _______.

(1) 1

(2) 3

(3) 0.5

(4) 2

In the following figure, *O* is the centre of the circle. If *∠**MPN *= 55°, then find the value of ∠*MON *+ *∠**OMN* + $\frac{1}{2}$*∠**MNO.*

(1) 145°

(2) $162\frac{1}{2}\xb0$

(3) $158\frac{1}{2}\xb0$

(4) 180°

In the following figure, Δ*PQR *is right-angled at *R* and *S* is the mid-point of hypotenuse *PQ*. If *RS *= 25 cm and *PR *= 48 cm, then find *QR*.

(1) 7 cm

(2) 25 cm

(3) 14 cm

(4) Cannot be determined

In a cyclic quadrilateral *PQRS, PS *= *PQ, RS* = *RQ* and ∠*PSQ* = 2∠*QSR. *Find ∠*QSR.*

(1) 20°

(2) 30°

(3) 40°

(4) 50°

In the following figure, two isosceles right triangles, *DEF* and *HGI* are on the same base $\overline{DH}$* *and $\overline{DH}$ is parallel to $\overline{FI}$. If *DE* = *GH* = 9 cm and *DH* = 20 cm, then the area of the quadrilateral *FEGI *is _______.

(1) 99 cm^{2}

(2) 40.5 cm^{2}

(3) 81 cm^{2}

(4) 180 cm^{2}

A pole of height 14 m casts a 10 m long shadow on the ground. At the same time, a tower casts a 70 m long shadow on the ground. Find the height of the tower.

(1) 50 m

(2) 78 m

(3) 90 m

(4) 98 m

The angle subtended by a minor arc in its alternate segment is _______.

(1) acute

(2) obtuse

(3) 90°

(4) reflex angle

The number of diagonals of a regular polygon is 27. Then, each of the interior angles of the polygon is _______.

(1) $\left(\frac{500}{3}\right)\xb0$

(2) 140°

(3) 128°

(4) 154°

*ABC *is a triangle inscribed in a circle, *AC *being the diameter of the circle. The length of *AC* is as much more than the length of *BC *as the length of *BC *is more than the length of *AB. *Find *AC : AB.*

(1) 5 : 3

(2) 5 : 4

(3) 6 : 5

(4) 3 : 2

$\overline{MN}$ is the arc of the circle with centre *O*. If ∠*MOR *= 100° and ∠*NOR *= 135°, then $\frac{1}{2}$∠O*RN + *$\frac{1}{4}$∠*ORM *is _______.

(1) $22\frac{1}{2}\xb0$

(2) 40°

(3) 125°

(4) $21\frac{1}{4}\xb0$

In the following figure (not to scale), ∠*BCD *= 40°, ∠*EDC *= 35°. ∠*CBF *= 30° and ∠*DEG *= 40°. Find ∠*BAE.*

(1) 70°

(2) 50°

(3) 110°

(4) 35°

In the following figure (not to scale), $\overline{AB}\parallel \overline{CD}$. If *∠**BAE *= 25° and *∠**DCE* = 30°, then find *∠**AEC*.

(1) 30°

(2) 45°

(3) 50°

(4) 35°

A tower of height 60 m casts a 40 m long shadow on the ground. At the same time, a needle of height 12 cm casts a *x *cm long shadow the ground. Find *x*.

(1) 6

(2) 8

(3) 10

(4) 14

In the given figure, *AC *is the diameter. *AB *and *AD *are equal chords. If ∠*AED *= 110°, then find ∠*BAD.*

(1) 40°

(2) 55°

(3) 110°

(4) 120°

In the given rectangle *ABCD, *the sum of the lengths of two diagonals is equal to 52 cm and *E *is a point in *AB, *such that $\overline{OE}$* *is perpendicular to $\overline{AB}$*. *Find the lengths of the sides of the rectangle, if *OE *= 5 cm.

(1) 24 cm, 10 cm

(2) 12 cm, 10 cm

(3) 24 cm, 5 cm

(4) 12 cm, 15 cm

In the following figure (not to scale), *AD *bisects ∠*BAC. *If ∠*BAD* = 45° and Δ*ABC* is inscribed in a circle, then which of the following is the longest?

(1) *AB*

(2) *AD*

(3) *AC*

(4) *BC*

In the given figure, $\overline{AB}\parallel \overline{DE}$* *and area of the parallelogram *ABFD* is 24 cm^{2}. Find the areas of Δ*AFB, ΔAGB *and* ΔAEB.*

(1) 8 cm^{2}

(2) 12 cm^{2}

(3) 10 cm^{2}

(4) 14 cm^{2}

In the given figure, $\overline{AD}$ and $\overline{BE}$ intersect at C, such that *BC *=* CE, **∠**ABC *= 40° and ∠*DEC *= 85°. Find ∠*BAC – *∠*CDE.*

(1) 45°

(2) 125°

(3) 55°

(4) 110°

In the given figure, *DEF* is a triangle. If *DF *is the longest side and *EF *is the shortest side, then which of the following is true?

(1) ∠*E *> ∠*D > *∠*F*

(2) ∠*D < *∠*F* *< *∠*E*

(3) ∠*D < *∠*E < *∠*F*

(4) None of these

The ratio between the exterior angle and the interior angle of a regular polygon is 1 : 3. Find the number of the sides of the polygon.

(1) 12

(2) 6

(3) 8

(4) 10

Find each interior and exterior angle of a regular polygon having 30 sides.

(1) 144°, 36°

(2) 156°, 24°

(3) 164°, 16°

(4) 168°, 12°

If *BC : CD* = 2 : 3, *AE : EC *= 3 : 4 and *BC : **AE *= 2 : 3, then find the ratio of the area of Δ*ECD *to the area of Δ*AEB.*

(1) 2 : 1

(2) 2 : 3

(3) 3 : 5

(4) 4 : 3

In the given figure (not to scale), *O* is the centre of the circle. If *PB *= *PC, *∠*PBO *= 25° and ∠BOC = 130°, then find ∠*ABP + *∠*DCP.*

(1) 10°

(2) 30°

(3) 40°

(4) 50°

In a polygon, the greatest angle is 110° and all the angles are distinct in integral measures (in degrees) Find the maximum number of sides it can have.

(1) 4

(2) 5

(3) 6

(4) 7

In the given figure, *ABCD *is a rectangle inscribed in a semi-circle. If the length and the breadth of the rectangle are in the ratio 2 : 1. What is the ratio of the perimeter of the rectangle to the diameter of the semicircle?

(1) 3 : $\sqrt{2}$

(2) $2:\sqrt{3}$

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

(4) $3:\sqrt{5}$

In the given figure, $\overline{AB}$* *the diameter of the circle with area π sq. units. Another circle is drawn with *C* as centre, which is on the given circle and passing through *A *and *B. *Find the area of the shaded region.

(1) $\frac{\pi}{3}$ sq. units

(2) $\frac{2\pi}{3}$ sq. units

(3) 1 sq. units

(4) 1.2 sq. units

In the following figure (not to scale). *C*_{1} and *C*_{2 }are two congruent circles with centres *O*_{1} and *O*_{2}, respectively. Each circle passes through the centre of the other circle. If the circumference of each circle is 2 cm, the perimeter of the shaded region is _______ cm.

(1) $\frac{4}{3}$

(2) 1

(3) $\frac{5}{3}$

(4) $\frac{2}{3}$

In the given figure (not to scale), the points *M, R, N, S* and *Q* are concyclic. Find ∠*PQR + *∠*OPR + *∠*NMS + *∠*OSN, *if *O* is the centre of the circle.

(1) 90°

(2) 180°

(3) 270°

(4) Data inadequate

In the given figure (not to scale), *AC *is the diameter of the circle and ∠*ADB *= 20°, then find ∠*BPC.*

(1) 50°

(2) 70°

(3) 90°

(4) 110°

In the following figure, *O* is the centre of the circle and *CD *=* DE*=* EF*= *GF. *If ∠*COD* = 40°, then find reflex ∠*COG.*

(1) 200°

(2) 90°

(3) 80°

(4) 160°

In the given figure (not to scale), *AC *is the median as well as altitude to *BD. *In Δ*ACE, AD *is the median to *CE. *Which of the following is true?

(1) *AB *+ *CD > AE*

(2) *AB* + *BC *=* AE*

(3) *AB + DE < AE*

(4) None of the above

In the given figure, (not to scale), rectangle *ABCD *and triangle *ABE *are inscribed in the circle with centre *O*. If ∠*AEB *= 40°, then find ∠*BOC.*

(1) 60°

(2) 80°

(3) 100°

(4) 120°

In the given figure (not to scale), *O* is the centre of the circle, *BC *and *CD *are equal chords. If ∠*OBC *= 55°, then find ∠*BAC.*

(1) 60°

(2) 70°

(3) 80°

(4) 90°

In the given figure (not to scale), *O* is the centre of the circle *C*_{1} and *AB *is the diameter of the circle *C*_{2}. Quadrilateral *PQRS *is inscribed in the circle with centre *O*. Find ∠*QRS.*

(1) 105°

(2) 115°

(3) 135°

(4) 145°

In the given figure (not to scale), *E *and D are the mid-points of *AB *and BC respectively. Also, ∠*B *= 90°, $AD=\sqrt{292}$ cm and $CM=\sqrt{208}$ cm. Find *AC.*

(1) 15

(2) 18

(3) 20

(4) 24

In Δ*ABC, P *is the mid-point of *BC* and *Q* is the mid-point of *AP*. Find the ratio of the area of Δ*ABQ* and the area of Δ*ABC. *The following are the steps involved in solving the above problem.

(A) We know that a median of a triangle divides a triangle into two triangles of equal area.

(B) $\Rightarrow \text{Ar}\left(\Delta ABP\right)=\frac{1}{2}\left[\text{Ar}\left(\Delta ABC\right)\right]$

(C) $\text{Ar}\left(\Delta ABQ\right)=\frac{1}{2}\left[\text{Ar}\left(\Delta ABP\right)\right]=\frac{1}{4}\left[\text{Ar}\Delta ABC\right]$

(D) $\Rightarrow \text{Ar}\left(\Delta ABQ\right):\text{Ar}\left(\Delta ABC\right)=1\text{\hspace{0.17em}\hspace{0.17em}}:\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}4$

(1) ACBD

(2) ADBC

(3) ABCD

(4) ADCB

*ABCD* is a cyclic quadrilateral, *ABC *is a minor arc and *O* is the centre of the circle. If ∠*AOC *= 160°, then find ∠*ABC.*

The following are the steps involved in solving the above problem. Arrange them in sequential order.

(A) We have, ∠*ABC + *∠*ADC *= 180°

(B) $\angle ABC+\frac{1}{2}\angle AOC={180}^{\circ}(\because \angle ADC=\frac{1}{2}\angle AOC)$

(C) ∠*ABC *= 180° – 80^{o}

(D) $\angle ABC+\frac{{160}^{\circ}}{2}={180}^{\circ}$

(E) ∴ ∠*ABC *= 100°

(1) ABDEC

(2) ABDCE

(3) BCDAE

(4) BACDE

Show that each diagonal of a parallelogram divide in into two congruent triangles.

The following are the steps involved in showing the above result. Arrange them in sequential order.

(A) In Δ*ABC* and Δ*CDA, AB* = *DC* and *BC* = *AD *($\because $ opposite angles of parallelogram) *AC* = *AC *(common side).

(B) Let *ABCD* be a parallelogram. Join *AC*.

(C) By SSS congruence property, $\Delta ABC\cong \Delta CDA.$

(D) Similarly, *BD* divides the triangle into two congruent triangles.

(1) BACD

(2) BDAC

(3) BADC

(4) BDCA

Show that any angle in a semi-circle is a right angle.

The following are the steps involved in showing the above result. Arrange them in sequential order.

(A) $\therefore \angle ACB=\frac{{180}^{\circ}}{2}={90}^{\circ}$

(B) The angle subtended by an are at the centre is double of the angle subtended by the same arc at any point on the remaining part of the circle.

(C) Let *AB* be a diameter of a circle with centre *D *and *C* be any point on the circle. Join *AC* and BC.

(D) ∴ ∠*ADB* = 2 × ∠*ACB*

180° = 2∠*ACB* (∵∠*ADB* = 180°)

(1) DBAC

(2) DBCA

(3) CBAD

(4) CBDA

*A, B, C,* and *D* are concyclic, AC bisects *BD.* If *AB *= 9 cm, *BC* = 8 cm, and *CD* = 6 cm, then find the measure of *AD*.

(1) 7 cm

(2) 10 cm

(3) 12 cm

(4) 15 cm

In the given figure, *PQRS* is a parallelogram, *A *and *B* are the mid points of $\overline{PQ}$* *and $\overline{SR}$* *respectively. If *PS* = *BR*, then the quadrilateral *ADBC *is a _______.

(1) rhombus

(2) trapezium

(3) square

(4) rectangle

The sides of a triangle are 2006 cm, 6002 cm and *m* cm, where *m *is a positive integer. Find the number of such possible triangles.

(1) 1

(2) 2006

(3) 3996

(4) 4011

If *a*, *b *and *c* are the lengths of the sides of a right triangle *ABC *with *c* = 2*a *and *b*^{2} – 3*a*^{2} = 0, then ∠*ABC *= _______.

(1) 60°

(2) 30°

(3) 45°

(4) 90°

In ∠*ABC, AC* = *BC, S *is the circum-centre and ∠*ASB* = 150°. Find ∠*CAB*.

(1) $55\frac{1}{2}\xb0$

(2) $52\frac{1}{2}\xb0$

(3) $62\frac{1}{2}\xb0$

(4) $35\frac{1}{2}\xb0$

In the given figure, *P, Q, R* and *S* are concyclic points, and *O* is the mid-point of the diameter *QS*.

If ∠*QPR *= 25°, then find ∠*SOR*.

(1) 130°

(2) 120°

(3) 75°

(4) 100°

In ΔABC, ∠B = 90°. *P, Q* and *R* are the mid-points of $\overline{AB}$*, *$\overline{BC}$ and $\overline{AC}$ respectively. Then which of the following is true?

(1) *A, P*, *Q* and *R* are concyclic points

(2) *B, P, R* and *Q* arc concyclic points

(3) *C, Q, P* and *R* are concyclic points

(4) All of these

If *p, q* and *r *are the lengths of the sides of a right triangle, *PQR*, and the hypotenuse $r=\sqrt{2pq}$*, *then ∠*QPR* = _______.

(1) 30°

(2) 45°

(3) 60°

(4) 30°

In a triangle *PQR**, PQ* =* QR. A *and *B *are the mid-points of $\overline{QR}$ and $\overline{PR}$ respectively. A circle passes through *P, Q, A* and *B. *Then which of the following is necessarily true?

(1) Δ*PQR* is equilateral

(2) ΔPQR is right isosceles

(3) *PQ *is a diameter

(4) Both (1) and (3)

In the figure given below (not to scale), *D* is a point on the circle with centre *A* and *C *is a point on the circle with centre *B. $\overline{AD}\perp \overline{BD}$ *and $\overline{BC}\perp \overline{CA}$*. *Then which of the following is true?

(1) *BD *=* AC, *when *AD *= *BC*

(2) *BD *=* AC, *when $\overline{AD}\parallel \overline{BC}$

(3) Both (1) and (2)

(4) *BD *= *AC* is always true

In the given figure, the angles ∠*ADE *and ∠*ABC *differ by 15°. Find ∠*CAE.*

(1) 10°

(2) $7\frac{1}{2}\xb0$

(3) 15°

(4) 30°

In the given figure, *ABCD *is a cyclic quadrilateral, ∠*ABC *= 70°, $\overline{FG}$ bisects ∠*CFA*, $\overline{EG}$ bisects *∠**DEB, **∠**DCE* = 60° and ∠*EGF *= 90°. Find ∠*HEC.*

(1) 20°

(2) 40°

(3) 25°

(4) 45°

In the given figure, *A, D, B, E *and *C* are concyclic If ∠*ACB* = 60° and ∠*AED *= 50°, then find ∠*DEB.*

(1) 15°

(2) 10°

(3) 20°

(4) 5°

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