Practice Questions for Geometry - Circles

1. PE 1 - Circles | Geometry - Circles

In the figure given below, CD = 30 cm and EF = 24 cm. Find the length of AB (in cm).

(a)
18
(b)
9
(c)
12
(d)
15

2. PE 1 - Circles | Geometry - Circles

In the given figure, ABC is a right angled triangle. ∠ABC = 90° and ∠ACB = 60°. If the radius of the smaller circle is 2 cm, then what is the radius (in cm) of the larger circle.

(a)
4
(b)
6
(c)
4.5
(d)
7.5

3. PE 1 - Circles | Geometry - Circles

AB and AC are the two tangents to a circle whose radius is 6 cm. If ∠BAC = 60°, then what is the value (in cm) of $\sqrt{A B^{2} + A C^{2}}$ ?

(a)
6√6
(b)
4√6
(c)
9√3
(d)
None of these

Answer: Option A

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Explanation :

OB = 6

∆ ABO is a 30°-60°-90° triangle

∴ AB = OB√3 = 6√3 cm

∴ $\sqrt{A B^{2} + A C^{2}}$ = $\sqrt{{(6 \sqrt{3})}^{2} + {(6 \sqrt{3})}^{2}}$ = 6√6 cm

Hence, option (a).

4. PE 1 - Circles | Geometry - Circles

In the given figure, O is the center of the circle. Circle has 3 tangents. If ∠QPR = 45°, then what is the value (in degrees) of ∠QOR?

(a)
67.5
(b)
72
(c)
78.5
(d)
65

5. PE 1 - Circles | Geometry - Circles

In the given figure, two identical circles of radius 4 cm touch each other. A and B are the centers of the two circles. If RQ and RS are tangents to the circle, then what is the length (in cm) of RQ?

(a)
3√3
(b)
2√6
(c)
4√2
(d)
6√2

Answer: Option C

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Explanation :

PQ = 4 + 4 + 4 + 4 = 16 cm

PA = 4 + 4 + 4 = 12 cm

In ∆ASR, PA² = AS² + PS²

⇒ 144 = 16 + PS²

⇒ PS = 8√2

Now, ∆PQR ~ ∆PSA

⇒ $\frac{R Q}{A S} = \frac{Q P}{P S}$

⇒ $\frac{R Q}{4} = \frac{16}{8 \sqrt{2}}$

⇒ RQ = 4√2 cm

Hence, option (c).

6. PE 1 - Circles | Geometry - Circles

The radius of two circles is 3 cm and 4 cm. The distance between their centers is 10 cm. What is the radio of the length of direct common tangent and transverse common tangent?

(a)
√51 : √68
(b)
√33 : √17
(c)
√66 : √51
(d)
√28 : √17

Answer: Option B

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Explanation :

Direct Common Tangent = $\sqrt{d^{2} - {(R - r)}^{2}}$ = $\sqrt{10^{2} - {(4 - 3)}^{2}}$ = √99

Transverse Common Tangent = $\sqrt{d^{2} - {(R + r)}^{2}}$ = $\sqrt{10^{2} - {(4 + 3)}^{2}}$ = √51

∴ Required ratio = √99 : √51 = √33 : √17

Hence, option (b).

7. PE 1 - Circles | Geometry - Circles

In the given figure, from the point P two tangents PA and PB are drawn to a circle with center O and radius 5 cm. From the point O, OC and OD are drawn parallel to PA and PB respectively. If the length of the chord AB is 5 cm, then what is the value (in degrees) of ∠COD?

(a)
90
(b)
120
(c)
150
(d)
135

8. PE 1 - Circles | Geometry - Circles

In the given figure, PQ is the diameter of the semicircle PABQ and O is its center. ∠AOB = 64°. BP cuts AQ at X. What is the value (in degrees) of ∠AXP?

(a)
36
(b)
58
(c)
32
(d)
54

9. PE 1 - Circles | Geometry - Circles

In the given figure, E and F are the centers of two identical circles. What is the ratio of area of triangle AOB to the area of triangle DOC?

(a)
1 : 3
(b)
1 : 9
(c)
1 : 8
(d)
1 : 4

Answer: Option B

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Explanation :

∆AEB ≅ MED

∴ MD = AB

∆BFA ≅ NFC

∴ NC = AB

⇒ DM = AB = NC

⇒ DC = DM + MN + NC = 3AB

Now, ∆AOB ≈ ∆COD

∴ $\frac{A r e a (∆ A O B)}{A r e a (∆ C O D)}$ = ${(\frac{1}{3})}^{2}$ = $\frac{1}{9}$

Hence, option (b).

10. PE 1 - Circles | Geometry - Circles

Two circles with radii 'a' & 'b' respectively touch each other externally. Let 'c' be the radius of a circle that touches these two circles as well as a common tangent to these two circles. Then which of the following is true?

(a)
a, b and c are in G.P.
(b)
a, b and c are in H.P.
(c)
1/c = 1/a + 1/b
(d)
$\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{a}}$

Answer: Option D

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Explanation :

Let C₁, C₂, & C₃ be three centers of three circles of radius a, b & c respectively PQ = 2 $\sqrt{a c}$ (Length of direct common tangent when two circles touch each other externally)

Similarly,

QR = 2 $\sqrt{b c}$ , PR = 2 $\sqrt{a b}$

PR = QR + PQ

2 $\sqrt{a b}$ = 2 $\sqrt{b c}$ + 2 $\sqrt{a c}$

Divide by √abc on both sides.

$\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{a}}$

Hence, option (d).

PE 1 - Circles | Geometry - Circles

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