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The radii of two circles are 4.5cm and 3.5 cm respectively. The distance between the centres of the circles is 10 cm. What is the length of the transverse common tangent?
- a)4 cm
- b)5 cm
- c)6 cm
- d)7 cm
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The radii of two circles are 4.5cm and 3.5 cm respectively. The distan...
In triangle OQS amd PRS
∠OQS = ∠PRS =90° [∵ Radius is perpendicular to tangent]
∠QSO = ∠PSR [∵ Vertically opposite angles are equal]
∴ OQS ∼ PRS
OQ/PR = OS/PS
4.5/3.5 = OS/PS
9/7 = OS/PS
∴ PS = 7OS/9
OP = OS + PS
10 = OS + 7OS/9
10 = 16OS/9
∴ OS = 90/16 = 5.625 cm
And PS = 10 – 5.625 = 4.375 cm
In triangle OQS, OS2 = OQ2 + QS2 [∵ Pythagoras theorem]
5.6252 = 4.52 + OS2
∴ OS = 3.375 cm
In triangle PRS, PS2 = PR2 + SP2 [∵ Pythagoras theorem]
4.3752 = 3.52 + SP2
∴ SP = 2.625
∴ OP = OS + SP
⇒ 3.375 + 2.625 = 6 cm
∠OQS = ∠PRS =90° [∵ Radius is perpendicular to tangent]
∠QSO = ∠PSR [∵ Vertically opposite angles are equal]
∴ OQS ∼ PRS
OQ/PR = OS/PS
4.5/3.5 = OS/PS
9/7 = OS/PS
∴ PS = 7OS/9
OP = OS + PS
10 = OS + 7OS/9
10 = 16OS/9
∴ OS = 90/16 = 5.625 cm
And PS = 10 – 5.625 = 4.375 cm
In triangle OQS, OS2 = OQ2 + QS2 [∵ Pythagoras theorem]
5.6252 = 4.52 + OS2
∴ OS = 3.375 cm
In triangle PRS, PS2 = PR2 + SP2 [∵ Pythagoras theorem]
4.3752 = 3.52 + SP2
∴ SP = 2.625
∴ OP = OS + SP
⇒ 3.375 + 2.625 = 6 cm
Most Upvoted Answer
The radii of two circles are 4.5cm and 3.5 cm respectively. The distan...
Given data:
- Radius of first circle = 4.5 cm
- Radius of second circle = 3.5 cm
- Distance between the centres of the circles = 10 cm
Calculating the length of transverse common tangent:
- Let's consider the two circles with radii R1 = 4.5 cm and R2 = 3.5 cm
- The distance between the centres of the circles is 10 cm
- The length of the transverse common tangent is given by the formula:
Length of transverse common tangent = √[(distance between centres)^2 - (R1 - R2)^2]
Plugging in the values:
- Length of transverse common tangent = √[(10)^2 - (4.5 - 3.5)^2]
- Length of transverse common tangent = √[100 - 1]
- Length of transverse common tangent = √99
- Length of transverse common tangent ≈ 9.95 cm
Therefore, the length of the transverse common tangent between the two circles is approximately 9.95 cm. Since this value is not provided in the options, the closest option is 6 cm (option 'C').
- Radius of first circle = 4.5 cm
- Radius of second circle = 3.5 cm
- Distance between the centres of the circles = 10 cm
Calculating the length of transverse common tangent:
- Let's consider the two circles with radii R1 = 4.5 cm and R2 = 3.5 cm
- The distance between the centres of the circles is 10 cm
- The length of the transverse common tangent is given by the formula:
Length of transverse common tangent = √[(distance between centres)^2 - (R1 - R2)^2]
Plugging in the values:
- Length of transverse common tangent = √[(10)^2 - (4.5 - 3.5)^2]
- Length of transverse common tangent = √[100 - 1]
- Length of transverse common tangent = √99
- Length of transverse common tangent ≈ 9.95 cm
Therefore, the length of the transverse common tangent between the two circles is approximately 9.95 cm. Since this value is not provided in the options, the closest option is 6 cm (option 'C').
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