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AC and BC are two equal chords of a circle. BA is produced to any point P and CP, when joined cuts the circle at T. Then (SSC CGL 1st Sit. 2012)
- a)CT : TP = AB : CA
- b)CT : TP = CA : AB
- c)CT : CB = CA : CP
- d)CT : CB = CP : CA
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
AC and BC are two equal chords of a circle. BA is produced to any poin...
Given:
- AC and BC are two equal chords of a circle.
- BA is produced to any point P.
- CP, when joined, cuts the circle at T.
We need to find the ratio CT:CB in terms of CA and CP.
Proof:
Let's assume that AB = BC = r (since AC and BC are equal chords).
1. Triangle ABC:
- Since AB = BC, triangle ABC is an isosceles triangle.
- Therefore, angle ABC = angle BAC.
2. Triangle ACP:
- Triangle ACP is a straight line.
- Therefore, angle BAC + angle ACP = 180 degrees.
3. Quadrilateral ABCT:
- Since angle ABC = angle BAC, angle BCT = 180 - 2(angle ABC).
- But angle ABC + angle ACP = 180 degrees.
- Therefore, angle BCT = 2(angle ABC).
4. Triangle BCT:
- In triangle BCT, angle BCT = 2(angle ABC).
- Therefore, angle BCT = 2(angle ABC) = 2(angle BAC).
5. Triangle BTC:
- Since angle BCT = 2(angle BAC), angle BTC = 180 - angle BCT.
- Therefore, angle BTC = 180 - 2(angle BAC).
6. Triangle CTP:
- In triangle CTP, angle CTP = angle BTC = 180 - 2(angle BAC).
7. Triangle CAT:
- In triangle CAT, angle CAT = angle CTP.
8. Triangle CAT and Triangle CAB:
- Triangle CAT and Triangle CAB are similar triangles (angle-angle similarity).
9. Using Similarity:
- By similarity, we can write the following ratio:
CT : CA = CP : CB
10. Rearranging the Ratio:
- Rearranging the above ratio, we get:
CT : CB = CA : CP
Therefore, the correct answer is option 'C': CT : CB = CA : CP.
- AC and BC are two equal chords of a circle.
- BA is produced to any point P.
- CP, when joined, cuts the circle at T.
We need to find the ratio CT:CB in terms of CA and CP.
Proof:
Let's assume that AB = BC = r (since AC and BC are equal chords).
1. Triangle ABC:
- Since AB = BC, triangle ABC is an isosceles triangle.
- Therefore, angle ABC = angle BAC.
2. Triangle ACP:
- Triangle ACP is a straight line.
- Therefore, angle BAC + angle ACP = 180 degrees.
3. Quadrilateral ABCT:
- Since angle ABC = angle BAC, angle BCT = 180 - 2(angle ABC).
- But angle ABC + angle ACP = 180 degrees.
- Therefore, angle BCT = 2(angle ABC).
4. Triangle BCT:
- In triangle BCT, angle BCT = 2(angle ABC).
- Therefore, angle BCT = 2(angle ABC) = 2(angle BAC).
5. Triangle BTC:
- Since angle BCT = 2(angle BAC), angle BTC = 180 - angle BCT.
- Therefore, angle BTC = 180 - 2(angle BAC).
6. Triangle CTP:
- In triangle CTP, angle CTP = angle BTC = 180 - 2(angle BAC).
7. Triangle CAT:
- In triangle CAT, angle CAT = angle CTP.
8. Triangle CAT and Triangle CAB:
- Triangle CAT and Triangle CAB are similar triangles (angle-angle similarity).
9. Using Similarity:
- By similarity, we can write the following ratio:
CT : CA = CP : CB
10. Rearranging the Ratio:
- Rearranging the above ratio, we get:
CT : CB = CA : CP
Therefore, the correct answer is option 'C': CT : CB = CA : CP.
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Community Answer
AC and BC are two equal chords of a circle. BA is produced to any poin...
In ΔPAC and ΔATC,
∠ATC = ∠PAC = 180° – θ.
∠PAC = ∠TCA
∴ ∠PAC ~ ΔATC
⇒ CT : CB = AC : PC
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