Theorem on Tangents MCQ Quiz in मल्याळम - Objective Question with Answer for Theorem on Tangents - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക
Last updated on Apr 15, 2025
Latest Theorem on Tangents MCQ Objective Questions
Top Theorem on Tangents MCQ Objective Questions
Theorem on Tangents Question 1:
In the following figure, the circles with centres B and D have radii 4 cm and x cm, respectively. AC is a tangent to both the circles. Find the value of x.
Answer (Detailed Solution Below)
Theorem on Tangents Question 1 Detailed Solution
Given:
Radius of small circle (r) = 4 cm
AE = 6 cm; EC = 9 cm
Concept used:
The tangent makes a right angle with the radius of a circle at the point of contact.
The vertical opposite angle is equal.
Calculation:
AD ⊥ AC and BC ⊥ AC
∠DAE = ∠BCE = 90°
In △DAE and △BCE
∠DAE = ∠BCE = 90°
∠AED = ∠BEC (Vertical angle)
△DAE ∼ △BCE (By AA similarity)
By C.P.C.T
⇒ AE/EC = AD/BC
⇒ 6/9 = 4/BC
⇒ BC = 36/6 = 6 cm
∴ The correct answer is 6 cm.
Theorem on Tangents Question 2:
Two circles with radii 22 cm and 16 cm touch each other externally. The length of the direct common tangent is:
Answer (Detailed Solution Below)
Theorem on Tangents Question 2 Detailed Solution
Given:
Radius (R) = 22 cm
radius (r) = 16 cm
Formula Used:
Length of common tangent = 2√(R × r)
R & r are radii
Calculation:
Solution:
Length of common tangent = 2√(22 × 16)
= 2√(352cm2)
= 2 × 4 √(22cm2)
= 8√(22)
Therefore, the length of the common tangent between the two circles is approximately 8√(22) cm.
Theorem on Tangents Question 3:
In the given figure, chords XY and PQ intersect each other at point L. Find the length of XY (in cm).
Answer (Detailed Solution Below)
Theorem on Tangents Question 3 Detailed Solution
Calculation
By the theorem,
LQ × LP = LY × LX
Let the length of XY be x.
⇒ 5 × 15 = 3 × (3 + x)
⇒ 25 = x + 3
⇒ x = 22
The length of XY is 22.
Theorem on Tangents Question 4:
In the given figure, if PA and PB are tangents to the circle with centre O such that ∠APB = 54°, then ∠OBA = ________.
Answer (Detailed Solution Below)
Theorem on Tangents Question 4 Detailed Solution
Given:
PA and PB are tangents to the circle with center O such that ∠APB = 54°
Calculation:
We know that the radius and tangent are perpendicular at their point of contact.
So,
∠PAO = ∠PBO = 90°
So,
∠AOB = 360 - 54 - 180
⇒ 126°
Also AO = OB = Radius
So, ∠OAB = ∠OBA = 54/2
⇒ 27°
∴ The required answer is 27°
Theorem on Tangents Question 5:
Two circles touch each other externally. The radius of the first circle with centre O is 6 cm. The radius of the second circle with centre P is 3 cm. Find the length of their common tangent AB.
Answer (Detailed Solution Below)
Theorem on Tangents Question 5 Detailed Solution
Given:
The radius of smaller circle = 3 cm
The radius of large circle = 6 cm
Formula used:
Direct common tangent = 2 × √(R × r)
Where, R = radius of large circle;
and r = radius of the small circle
Calculation:
Direct common tangent = 2 × √(R × r)
⇒ 2 × √(3 × 6)
⇒ 2 × 3 × √2 = 6√2 cm
∴ The correct answer is 6√2 cm.
Theorem on Tangents Question 6:
If PT is a tangent at T to a circle whose centre is O and OP = 17 cm and OT = 8 cm, find the length of the tangent segment PT
Answer (Detailed Solution Below)
Theorem on Tangents Question 6 Detailed Solution
Given:
PT is a tangent to a circle, whose centre is O.
OP = 17 cm
OT = 8 cm
Formula used:
Pythagoras Theorem: In a right-angle triangle, the hypotenuse is equal to -
OP2 = OT2 + TP2
Calculation:
This, the length of the tangent TP is equal to:
TP2 = OP2 - OT2
TP2 = (17)2 - (8)2
TP2 = 289 - 64
TP2 = 225
TP = √225 = 15
∴ The length of the tangent segment PT is equal to 15 cm.
Theorem on Tangents Question 7:
The diameters of two circles are 12 cm and 20 cm, respectively and the distance between their centres is 16 cm. Find the number of common tangents to the circles.
Answer (Detailed Solution Below)
Theorem on Tangents Question 7 Detailed Solution
Given:
The diameters of the two circles are 12 cm and 20 cm
The distance between their centres is 16 cm
Calculation:
According to the diagram,
The two circles are touching each other externally
So, the number of common tangents will be 3
∴ The required answer is 3.
Theorem on Tangents Question 8:
Two circles of radii 18 cm and 12 cm touch each other externally. Find the length (in cm) of their direct common tangent.
Answer (Detailed Solution Below)
Theorem on Tangents Question 8 Detailed Solution
Shortcut Trick
When two circles touch each other externally then the length of their direct common tangent = 2√(r1 × r2)
So, the length of the common tangent
= 2 √(18 × 12)
= 2 √(6 × 3 × 6 × 2)
= 2 × 6 √6 = 12√6 cm
∴ The correct answer is option (3).
Alternate Method
Given:
r1 = 18 cm and r2 = 12 cm
Concept:
The length of the direct common tangent = √(d2 - (r1 - r2)2)
Where: d = distance between the centres of the circles
r1 and r2 are the radii of the circles.
Calculation:
d = r1 + r2 = 18 + 12 = 30 cm
So,
The length of the direct common tangent = √(d2 - (r1 - r2)2)
⇒ √(302 - (18 - 12)2)
⇒ √(302 - 62)
⇒ √{(30 + 6) (30 - 6)}
⇒ √(36 × 24)
⇒ √(36 × 4 × 6)
⇒ (6 × 2)√6
⇒ 12√6 cm
∴ The correct answer is option (3).
Theorem on Tangents Question 9:
A secant PAB is drawn from an external point P to the circle with the centre at O, intersecting it at A and B. If OP = 17 cm, PA = 12 cm and PB = 22.5 cm, then the radius of the circle is:
Answer (Detailed Solution Below)
Theorem on Tangents Question 9 Detailed Solution
Given:
OP = 17 cm
PA = 12 cm
PB = 22.5 cm
Formula Used:
In such a case = PA × PB = PC × PD
Calculation:
Let radius = x
⇒ PC = 17 - x
and PD = 17 + x
According to Question
PA × PB = PC × PD
⇒ 12 × 22.5 = (17 - x)(17 + x)
⇒ 270 = 289 - x2
⇒ x2 = 19
⇒ x = √19
⇒ r = √(19) cm
The radius of the circle is √(19) cm.
Theorem on Tangents Question 10:
AB is a chord of length 32 cm of a circle of radius 20 cm, the tangents at A and B intersect at a point T. Find length TA (rounded off to two digits after decimal).
Answer (Detailed Solution Below)
Theorem on Tangents Question 10 Detailed Solution
Given:
Length of chord AB = 32 cm.
Radius of the circle = 20 cm.
Calculation:
In triangle OBC:
tan BOC = (BC/OC) = 16/12 = 4/3
Triangle OTB:
tan TOB = (BT/OB)
4/3 = BT/ 20
TB = 80/3 = 26.66
Hence,
TA = 26.67 cm
The correct answer is 26.67 cm