Question

Four alternative answers for each of the following questions are given. Choose the correct alternative.
(1) Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between their centers ?

(A) 4.4 cm (B) 8.8 cm (C) 2.2 cm (D) 8.8 or 2.2 cm

(2) Two circles intersect each other such that each circle passes through the centre of the other. If the distance between their centres is 12, what is the radius of each circle ?

(A) 6 cm (B) 12 cm (C) 24 cm (D) can’t say

(3) A circle touches all sides of a parallelogram. So the parallelogram must be a, ................... .

(A) rectangle (B) rhombus (C) square (D) trapezium

(4) Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.

(A) 25 cm (B) 24 cm (C) 7 cm (D) 14 cm

(5) If two circles are touching externally, how many common tangents of them can be drawn?

(A) One (B) Two (C) Three (D) Four

(6) ∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB).

(A) 65° (B) 130° (C) 295° (D) 230°

(7) Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6, EB = 10, CE = 8, find ED.

(A) 7 (B) 8 (C) 11.2 (D) 9

(8) In a cyclic ▢ABCD, twice the measure of ∠A is thrice the measure of ∠C. Find the measure of ∠C?

(A) 36 (B) 72 (C) 90 (D) 108

(9) Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No point, except point B, is common to the arcs. Which is the type of ∆ABC?

(A) Equilateral triangle (B) Scalene triangle (C) Right angled triangle (D) Isosceles triangle

(10) Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ is an acute angle. (ii) ∠XYZ can’t be a right angle. (iii) ∠XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of ∠XYZ.

(A) Only one (B) Only two (C) Only three (D) All

Answer 1

(1)
The radii of the two circles are 5.5 cm and 3.3 cm.

If two circles touch each other externally, distance between their centres is equal to the sum of their radii.

∴ Distance between their centres = 5.5 cm + 3.3 cm = 8.8 cm

If two circles touch each other internally, distance between their centres is equal to the difference of their radii.

∴ Distance between their centres = 5.5 cm − 3.3 cm = 2.2 cm

Thus, the distance between their centres is 8.8 cm or 2.2 cm

Hence, the correct answer is option (D).

(2)
Let C₁ and C₂ be the centres of the two circles.



Radius of circle with centre C₁ = Radius of circle with centre C₂ = Distance between their centres = C₁C₂ = 12 cm

Thus, the radius of each circle is 12 cm.

Hence, the correct answer is option (B).

(3)
ABCD is a parallelogram. A circle with centre O touches the parallelogram at E, F, G and H.



ABCD is a parallelogram.

∴ AB = CD .....(1) (Opposite sides of parallelogram are equal)

AD = BC .....(2) (Opposite sides of parallelogram are equal)

Tangent segments drawn from an external point to a circle are congruent.

AE = AH .....(3)

DG = DH .....(4)

BE = BF .....(5)

CG = CF .....(6)

Adding (3), (4), (5) and (6), we get

AE + BE + CG + DG = AH + DH + BF + CF

⇒ AB + CD = AD + BC .....(7)

From (1), (2) and (7), we ahve

2AB = 2BC

⇒ AB = BC .....(8)

From (1), (2) and (8), we have

AB = BC = CD = AD

∴ Parallelogram ABCD is a rhombus. (A rhombus is a parallelogram with all sides equal)

Hence, the correct answer is option (B).

(4)
Let O be the centre of the circle and AB be the tangent segment drawn from an external point A touching the circle at B.



The tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ ∠ABO = 90º

In right ∆ABO,

$$\begin{gathered} {\mathrm{OA}}^2={\mathrm{AB}}^2+{\mathrm{OB}}^2 \\ \Rightarrow \mathrm{OB}=\sqrt{{\mathrm{OA}}^2-{\mathrm{AB}}^2} \\ \Rightarrow \mathrm{OB}=\sqrt{{\left(12.5\right)}^2-{\left(12\right)}^2} \\ \Rightarrow \mathrm{OB}=\sqrt{156.25-144} \\ \Rightarrow \mathrm{OB}=\sqrt{12.25}=3.5\mathrm{cm} \end{gathered}$$
Radius of the circle = OB = 3.5 cm

∴ Diameter of the circle = 2 × Radius of the circle = 2 × 3.5 = 7 cm

Hence, the correct answer is option (C).

(5)
If two circles touch each other externally, then three common tangents can drawn to the circles.



Hence, the correct answer is option (C).

(6)


The measure of an inscribed angle is half of the measure of the arc intercepted by it.

∴∠ACB = $\frac{1}{2}$ m(arc AB)

⇒ m(arc AB) = 2∠ACB = 2 × 65º = 130º

∴ m(arc ACB) = 360º − m(arc AB) = 360º − 130º = 230º

Hence, the correct answer is option (D).

(7)


If two chords of a circle intersect each other in the interior of the circle, then the product of the lengths of the two tangents of one chord is equal to the product of the lengths of the two segments of the other chord.

∴ AE × EB = CE × ED

⇒ 5.6 × 10 = 8 × ED

⇒ ED = $\frac{56}{8}$ = 7 units

Hence, the correct answer is option (A).

(8)
ABCD is a cyclic quadrilateral.

2∠A = 3∠C .....(1) (Given)

Now,

∠A + ∠C = 180º (Opposite angles of a cyclic quadrilateral are supplementary)

⇒ $\frac{3}{2}$∠C + ∠C = 180º [From (1)]

​⇒ $\frac{5}{2}$∠C = 180º

⇒ ∠C = $\frac{2\times 180°}{5}$ = 72º

Thus, the measure of ∠C is 72º.

Hence, the correct answer is option (B).

(9)


m(arc AB) = m(arc BC) = 120º

Now,

m(arc AB) + m(arc BC) + m(arc CA) = 360º

⇒ 120º + 120º + m(arc CA) = 360º

⇒ 240º + m(arc CA) = 360º

⇒ m(arc CA) = 360º − 240º = 120º

∴ m(arc AB) = m(arc BC) = m(arc CA)

⇒ arc AB ≅ arc BC ≅ arc CA (Two arcs are congruent if their measures are equal)

⇒ chord AB ≅ chord BC ≅ chord CA (Chords corresponding to congruent arcs of a circle are congruent)

∴ ∆ABC is an equilateral triangle. (All sides of equilateral triangle are equal)

Hence, the correct answer is option (A).

(10)
Let P be any point on the arc XZ.



XZ is the diameter of the circle.

∴ ∠XPZ = 90º (Angle in a semi-circle is 90º)

So, ∠XYZ cannot be a right angle.

In ∆YPZ,

∠XYZ > ​∠YPZ (An exterior angle of a triangle is greater than its remote interior angle)

⇒ ∠XYZ > ​90º (∠YPZ = ∠XPZ)

So, ∠XYZ is an obtuse angle. Therefore, it is not possible that ∠XYZ is an acute angle.

Thus, three of the following statements are true.

Hence, the correct answer is option (C).