CAT Exam > CAT Questions > On a triangle ABC, a circle with diameter BC...
Start Learning for Free
On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, is
Correct answer is '24'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
On a triangle ABC, a circle with diameter BC is drawn, intersecting A...
Let us draw the diagram according to the available information.
View all questions of this test
We can see that triangle BPC and BQC are inscribed inside a semicircle. Hence, we can say that
Therefore, we can say that BQ ⊥ AC and CP ⊥ AB.
In triangle ABC,
Area of triangle = (1/2)*Base*Height = (1/2)*AB*CP = (1/2)*AC*BQ
=> BQ = AB *CP/AC = 30 * 20/25 = 24 cm.
Most Upvoted Answer
On a triangle ABC, a circle with diameter BC is drawn, intersecting A...
To find the length of BQ, we can use the property of circles that states that angles formed by intersecting lines and a circle are equal. Let's break down the solution into steps:
1. Draw the triangle ABC with AB = 30 cm, AC = 25 cm, and CP = 20 cm.
2. Draw the circle with diameter BC. Let's label the center of the circle as O.
3. Let's label the points of intersection of the circle with AB and AC as P and Q, respectively.
4. Since the circle has diameter BC, angle BOC is a right angle.
5. Let's denote the length of BQ as x.
6. Using the property mentioned earlier, we can say that angle BPC is equal to angle BQC.
7. Since angle BOC is a right angle, angles BPC and BQC are both 90 degrees.
8. This means that triangle BPQ is a right-angled triangle with BP as the hypotenuse.
9. Using the Pythagorean theorem, we can find the length of BP:
- BP^2 = BQ^2 + PQ^2
- BP^2 = x^2 + (30 - x)^2 [Since BP = AB - AP = 30 - x]
- BP^2 = x^2 + 900 - 60x + x^2
- BP^2 = 2x^2 - 60x + 900
10. We also know that triangle CPQ is a right-angled triangle with CP as the hypotenuse.
11. Using the Pythagorean theorem, we can find the length of CP:
- CP^2 = CQ^2 + PQ^2
- CP^2 = (25 - x)^2 + x^2 [Since CP = AC - AQ = 25 - x]
- CP^2 = 625 - 50x + x^2 + x^2
- CP^2 = 2x^2 - 50x + 625
12. Since CP = 20 cm, we can equate the expressions for CP^2 from steps 11 and 12:
- 2x^2 - 50x + 625 = 400
- 2x^2 - 50x + 225 = 0
13. Solving this quadratic equation, we find that x = 24 or x = 4.5.
- We discard x = 4.5 since it is not a feasible length for BQ.
14. Therefore, the length of BQ is 24 cm, as given in the correct answer.
1. Draw the triangle ABC with AB = 30 cm, AC = 25 cm, and CP = 20 cm.
2. Draw the circle with diameter BC. Let's label the center of the circle as O.
3. Let's label the points of intersection of the circle with AB and AC as P and Q, respectively.
4. Since the circle has diameter BC, angle BOC is a right angle.
5. Let's denote the length of BQ as x.
6. Using the property mentioned earlier, we can say that angle BPC is equal to angle BQC.
7. Since angle BOC is a right angle, angles BPC and BQC are both 90 degrees.
8. This means that triangle BPQ is a right-angled triangle with BP as the hypotenuse.
9. Using the Pythagorean theorem, we can find the length of BP:
- BP^2 = BQ^2 + PQ^2
- BP^2 = x^2 + (30 - x)^2 [Since BP = AB - AP = 30 - x]
- BP^2 = x^2 + 900 - 60x + x^2
- BP^2 = 2x^2 - 60x + 900
10. We also know that triangle CPQ is a right-angled triangle with CP as the hypotenuse.
11. Using the Pythagorean theorem, we can find the length of CP:
- CP^2 = CQ^2 + PQ^2
- CP^2 = (25 - x)^2 + x^2 [Since CP = AC - AQ = 25 - x]
- CP^2 = 625 - 50x + x^2 + x^2
- CP^2 = 2x^2 - 50x + 625
12. Since CP = 20 cm, we can equate the expressions for CP^2 from steps 11 and 12:
- 2x^2 - 50x + 625 = 400
- 2x^2 - 50x + 225 = 0
13. Solving this quadratic equation, we find that x = 24 or x = 4.5.
- We discard x = 4.5 since it is not a feasible length for BQ.
14. Therefore, the length of BQ is 24 cm, as given in the correct answer.
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer?
Question Description
On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. 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 On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer?.
On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. 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 On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer?.
Solutions for On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer?, a detailed solution for On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? has been provided alongside types of On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, isCorrect answer is '24'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup