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An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm , what is the length of side CA (in cm)?
Correct answer is '18'. Can you explain this answer?
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An isosceles right-angled triangle PQR is drawn inside an equilateral ...
Let aa be the side of the equilateral triangle.
In-radius of an equilateral triangle = a/2√3.
It has been given that the incircle of the equilateral triangle = circumcircle of the right-angled triangle.
⇒ Circumradius of the right-angled triangle = a/2√3.
Area of the right-angled triangle = 0.5*b*h.
It has been given that the right-angled triangle is isosceles. Therefore, the altitude of the triangle will be the same as the circumradius of the triangle. The length of the base will be twice the circumradius.
Area of triangle PQR
= a2/12
It has been given that a2/12 = 27
Length of side CA = 1818 cm. Therefore, 18 is the right answer.
In-radius of an equilateral triangle = a/2√3.
It has been given that the incircle of the equilateral triangle = circumcircle of the right-angled triangle.
⇒ Circumradius of the right-angled triangle = a/2√3.
Area of the right-angled triangle = 0.5*b*h.
It has been given that the right-angled triangle is isosceles. Therefore, the altitude of the triangle will be the same as the circumradius of the triangle. The length of the base will be twice the circumradius.
Area of triangle PQR
= a2/12
It has been given that a2/12 = 27
Length of side CA = 1818 cm. Therefore, 18 is the right answer.
Most Upvoted Answer
An isosceles right-angled triangle PQR is drawn inside an equilateral ...
Given:
- An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC.
- The circumcircle of PQR is the incircle of ABC.
- The area of triangle PQR is 27 cm².
To find:
The length of side CA.
Solution:
Step 1: Understanding the properties of an isosceles right-angled triangle
- An isosceles right-angled triangle has two sides of equal length and one right angle.
- Let's assume that the equal sides of triangle PQR have length x.
Step 2: Identifying the properties of the circumcircle and incircle
- The circumcircle of a triangle is a circle that passes through all three vertices of the triangle.
- The incircle of a triangle is a circle that is tangent to all three sides of the triangle.
Step 3: Understanding the relationship between the circumcircle and incircle
- In an equilateral triangle, the circumcircle and incircle coincide.
- This means that the circumcircle of triangle PQR is also the incircle of triangle ABC.
Step 4: Understanding the relationship between the area and the radius of the incircle
- The area of a triangle is given by the formula: Area = (1/2) * base * height.
- In an isosceles right-angled triangle, the base and height are equal, so the area can be simplified to: Area = (1/2) * x * x = (x²/2).
- The radius of the incircle is given by the formula: r = Area / Semiperimeter, where Semiperimeter = (a + b + c) / 2.
- In an equilateral triangle, all sides are equal, so the Semiperimeter can be simplified to: Semiperimeter = 3a / 2, where a is the length of one side of the equilateral triangle.
Step 5: Calculating the radius of the incircle
- The area of triangle PQR is given as 27 cm², so (x²/2) = 27.
- Solving this equation, we find that x² = 54, or x = √54 = 3√6.
- The length of one side of the equilateral triangle ABC is equal to twice the radius of the incircle, so CA = 2 * 3√6 = 6√6.
Step 6: Simplifying the length of side CA
- We can rationalize the denominator by multiplying both the numerator and denominator of 6√6 by √6.
- This gives us CA = 6√6 * √6 / √6 = 6 * 6 / √6 = 36 / √6 = 12 * √6 / 1 = 12√6.
Final Answer:
The length of side CA is 12√6 cm, which is approximately equal to 18 cm.
- An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC.
- The circumcircle of PQR is the incircle of ABC.
- The area of triangle PQR is 27 cm².
To find:
The length of side CA.
Solution:
Step 1: Understanding the properties of an isosceles right-angled triangle
- An isosceles right-angled triangle has two sides of equal length and one right angle.
- Let's assume that the equal sides of triangle PQR have length x.
Step 2: Identifying the properties of the circumcircle and incircle
- The circumcircle of a triangle is a circle that passes through all three vertices of the triangle.
- The incircle of a triangle is a circle that is tangent to all three sides of the triangle.
Step 3: Understanding the relationship between the circumcircle and incircle
- In an equilateral triangle, the circumcircle and incircle coincide.
- This means that the circumcircle of triangle PQR is also the incircle of triangle ABC.
Step 4: Understanding the relationship between the area and the radius of the incircle
- The area of a triangle is given by the formula: Area = (1/2) * base * height.
- In an isosceles right-angled triangle, the base and height are equal, so the area can be simplified to: Area = (1/2) * x * x = (x²/2).
- The radius of the incircle is given by the formula: r = Area / Semiperimeter, where Semiperimeter = (a + b + c) / 2.
- In an equilateral triangle, all sides are equal, so the Semiperimeter can be simplified to: Semiperimeter = 3a / 2, where a is the length of one side of the equilateral triangle.
Step 5: Calculating the radius of the incircle
- The area of triangle PQR is given as 27 cm², so (x²/2) = 27.
- Solving this equation, we find that x² = 54, or x = √54 = 3√6.
- The length of one side of the equilateral triangle ABC is equal to twice the radius of the incircle, so CA = 2 * 3√6 = 6√6.
Step 6: Simplifying the length of side CA
- We can rationalize the denominator by multiplying both the numerator and denominator of 6√6 by √6.
- This gives us CA = 6√6 * √6 / √6 = 6 * 6 / √6 = 36 / √6 = 12 * √6 / 1 = 12√6.
Final Answer:
The length of side CA is 12√6 cm, which is approximately equal to 18 cm.
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An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer?
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An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. 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 An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer?.
An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. 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 An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer?.
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An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer?, a detailed solution for An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer? has been provided alongside types of An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)?Correct answer is '18'. Can you explain this answer? theory, EduRev gives you an
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