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Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:
- a)PQ/3
- b)PQ/6
- c)PQ/√2
- d)PQ/4
- e)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Semicircle C1 is drawn with a line segment PQ as its diameter with ce...
To find the radius of circle C4, we can use the concept of tangents and the properties of circles.
Properties of Tangents:
1. A line that touches a circle at only one point is called a tangent.
2. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Properties of Circles:
1. The perpendicular from the center of a circle to a chord bisects the chord.
2. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Step 1: Draw the diagram
Draw semicircles C1, C2, and C3 inside C1 with diameter PQ, PR, and QR respectively. Draw the circle C4 tangent to all three semicircles.
Step 2: Identify the given information
We are given that PQ is the diameter of the large semicircle C1 and that C2 and C3 lie inside C1. We need to find the radius of circle C4.
Step 3: Determine the key points
Let O be the center of circle C4. Let M, N, and S be the points of tangency between circle C4 and semicircles C1, C2, and C3 respectively. Let T be the center of semicircle C1.
Step 4: Use the properties of tangents
Since circle C4 is tangent to semicircle C1 at point M, we know that TM is perpendicular to MQ (the radius of C1). Similarly, since circle C4 is tangent to semicircle C2 at point N, TN is perpendicular to NP (the radius of C2). Finally, since circle C4 is tangent to semicircle C3 at point S, TS is perpendicular to SQ (the radius of C3).
Step 5: Identify the key relationships
Since MT, NT, and TS are all perpendicular to the radii MQ, NP, and SQ respectively, they are all equal to the radius of circle C4.
Step 6: Determine the equation
We can use the relationship between the radii to find the radius of C4. Since MQ = NP + NQ and TS = SQ + ST, we have:
MQ = NP + NQ = NP + QR/2
TS = SQ + ST = SQ + PR/2
We know that MQ = TS, so:
NP + QR/2 = SQ + PR/2
Since NP = PR/2 and QR = SQ/2 (as they are diameters of semicircles), we can substitute these values into the equation:
PR/2 + SQ/4 = SQ + PR/2
Simplifying the equation, we get:
PR/2 = 3SQ/4
Multiplying both sides by 4/3, we have:
PR = 2SQ/3
Since PR = PQ and SQ = QR (as they are diameters of semicircles), we can substitute these values into the equation:
PQ = 2QR/3
Multiplying both sides by 3/2, we have:
QR = PQ/2
Hence, the radius of circle C4 is PQ/2 or PQ/6.
Therefore, the correct answer is
Properties of Tangents:
1. A line that touches a circle at only one point is called a tangent.
2. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Properties of Circles:
1. The perpendicular from the center of a circle to a chord bisects the chord.
2. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Step 1: Draw the diagram
Draw semicircles C1, C2, and C3 inside C1 with diameter PQ, PR, and QR respectively. Draw the circle C4 tangent to all three semicircles.
Step 2: Identify the given information
We are given that PQ is the diameter of the large semicircle C1 and that C2 and C3 lie inside C1. We need to find the radius of circle C4.
Step 3: Determine the key points
Let O be the center of circle C4. Let M, N, and S be the points of tangency between circle C4 and semicircles C1, C2, and C3 respectively. Let T be the center of semicircle C1.
Step 4: Use the properties of tangents
Since circle C4 is tangent to semicircle C1 at point M, we know that TM is perpendicular to MQ (the radius of C1). Similarly, since circle C4 is tangent to semicircle C2 at point N, TN is perpendicular to NP (the radius of C2). Finally, since circle C4 is tangent to semicircle C3 at point S, TS is perpendicular to SQ (the radius of C3).
Step 5: Identify the key relationships
Since MT, NT, and TS are all perpendicular to the radii MQ, NP, and SQ respectively, they are all equal to the radius of circle C4.
Step 6: Determine the equation
We can use the relationship between the radii to find the radius of C4. Since MQ = NP + NQ and TS = SQ + ST, we have:
MQ = NP + NQ = NP + QR/2
TS = SQ + ST = SQ + PR/2
We know that MQ = TS, so:
NP + QR/2 = SQ + PR/2
Since NP = PR/2 and QR = SQ/2 (as they are diameters of semicircles), we can substitute these values into the equation:
PR/2 + SQ/4 = SQ + PR/2
Simplifying the equation, we get:
PR/2 = 3SQ/4
Multiplying both sides by 4/3, we have:
PR = 2SQ/3
Since PR = PQ and SQ = QR (as they are diameters of semicircles), we can substitute these values into the equation:
PQ = 2QR/3
Multiplying both sides by 3/2, we have:
QR = PQ/2
Hence, the radius of circle C4 is PQ/2 or PQ/6.
Therefore, the correct answer is
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Community Answer
Semicircle C1 is drawn with a line segment PQ as its diameter with ce...
Let radius of C4 be r
Let PQ be d. We have PR = d/2 = RQ = RO
So, RS = d/2 - r, RT = d/4, ST = d/4 + r
In right triangle STR,
RS2 + RT2 = ST2
(d/4 + r)2 = (d/4)2 + (d/2 - r)2
Solving, r = d/6 = PQ/6
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Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer?
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Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer?.
Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Semicircle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as diameters respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside C2 and C3. The radius of C4 is:a)PQ/3b)PQ/6c)PQ/√2d)PQ/4e)None of theseCorrect answer is option 'B'. Can you explain this answer?.
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