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The difference between the lengths of the inradius and circumradius of a right-angled triangle is equal to a fourth of the length of the hypotenuse of the triangle. A square is constructed using the hypotenuse of the triangle as the base. What is the ratio of the areas of the triangle and the square?
- a)1:2
- b)1:3
- c)7:16
- d)5:16
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The difference between the lengths of the inradius and circumradius of...
We know that the length of the inradius of a right-angled triangle is given by the formula (a+b-h)/2 , where aa and bb are the lengths of the arms of the triangle and hh is the length of the hypotenuse of the triangle.
Also, we know that the length of the circumradius of a right-angled triangle is h/2.
Circumradius of a triangle is always greater than its inradius.
Squaring on both sides, we get,
We know that in a right-angled triangle, a2 + b2 = h2.
Substituting in (1), we get,
We know that the area of the triangle is ab/2.
Area of the square = h2.
Ratio of the areas of the triangle and the square
Therefore, option D is the right answer.
Alternate solution:
Let 'R' be the circumradius of the triangle and 'r' be the inradius.
For a right-angled triangle, area = r(r+2R)
It has been given that difference between the lengths of circumradius and inradius is equal to one-fourth the hypotenuse of the triangle.
Hypotenuse = 2*circumradius
⇒ R - r = R/2
R = 2r
Area of the triangle = r(r + 2 * 2r)= 5r2
Area of the square = (2R)2 = (2*2r)2 = 16r2
Therefore, the required ratio is 5:16.
Also, we know that the length of the circumradius of a right-angled triangle is h/2.
Circumradius of a triangle is always greater than its inradius.
Squaring on both sides, we get,
We know that in a right-angled triangle, a2 + b2 = h2.
Substituting in (1), we get,
We know that the area of the triangle is ab/2.
Area of the square = h2.
Ratio of the areas of the triangle and the square
Therefore, option D is the right answer.
Alternate solution:
Let 'R' be the circumradius of the triangle and 'r' be the inradius.
For a right-angled triangle, area = r(r+2R)
It has been given that difference between the lengths of circumradius and inradius is equal to one-fourth the hypotenuse of the triangle.
Hypotenuse = 2*circumradius
⇒ R - r = R/2
R = 2r
Area of the triangle = r(r + 2 * 2r)= 5r2
Area of the square = (2R)2 = (2*2r)2 = 16r2
Therefore, the required ratio is 5:16.
This question is part of UPSC exam. View all CAT courses
Most Upvoted Answer
The difference between the lengths of the inradius and circumradius of...
Given:
- The difference between the lengths of the inradius and circumradius of a right-angled triangle is equal to a fourth of the length of the hypotenuse of the triangle.
- A square is constructed using the hypotenuse of the triangle as the base.
To find:
The ratio of the areas of the triangle and the square.
Solution:
Let's assume the sides of the right-angled triangle are a, b, and c, where c is the hypotenuse.
We know that the inradius of a right-angled triangle is given by r = (a + b - c)/2.
Difference between the inradius and circumradius:
The circumradius of a right-angled triangle is given by R = c/2.
Given that the difference between the inradius and circumradius is equal to a fourth of the length of the hypotenuse:
r - R = c/4
(a + b - c)/2 - c/2 = c/4
(a + b - c) - c = c/2
a + b - 2c = c/2
a + b = (5/2)c
Area of the triangle:
The area of a right-angled triangle is given by A = (1/2) * a * b.
Area of the square:
The side of the square is equal to the hypotenuse of the triangle, so the area of the square is given by A' = c^2.
Ratio of the areas:
The ratio of the areas of the triangle and the square is given by A/A':
(A)/(c^2) = (1/2) * a * b / c^2
We know that a + b = (5/2)c, so substituting this value:
(A)/(c^2) = (1/2) * (a + b) * (a - b) / c^2
(A)/(c^2) = (1/2) * (5/2)c * (a - b) / c^2
(A)/(c^2) = (5/4) * (a - b) / c
We also know that a^2 + b^2 = c^2 (Pythagoras theorem), so substituting this value:
(A)/(c^2) = (5/4) * (a - b) / c
(A)/(c^2) = (5/4) * (sqrt(c^2 - b^2) - b) / c
Simplifying further:
(A)/(c^2) = (5/4) * (sqrt(c^2 - b^2)/c - b/c)
(A)/(c^2) = (5/4) * (sqrt(1 - (b/c)^2) - b/c)
Since b/c is the sine of the angle opposite to side b, let's denote it as sin(B):
(A)/(c^2) = (5/4) * (sqrt(1 - sin^2(B)) - sin(B))
Using the trigonometric identity sin^2(B) + cos^2(B) = 1, we can write the ratio as:
(A)/(c^2) = (5/4) * (cos(B
- The difference between the lengths of the inradius and circumradius of a right-angled triangle is equal to a fourth of the length of the hypotenuse of the triangle.
- A square is constructed using the hypotenuse of the triangle as the base.
To find:
The ratio of the areas of the triangle and the square.
Solution:
Let's assume the sides of the right-angled triangle are a, b, and c, where c is the hypotenuse.
We know that the inradius of a right-angled triangle is given by r = (a + b - c)/2.
Difference between the inradius and circumradius:
The circumradius of a right-angled triangle is given by R = c/2.
Given that the difference between the inradius and circumradius is equal to a fourth of the length of the hypotenuse:
r - R = c/4
(a + b - c)/2 - c/2 = c/4
(a + b - c) - c = c/2
a + b - 2c = c/2
a + b = (5/2)c
Area of the triangle:
The area of a right-angled triangle is given by A = (1/2) * a * b.
Area of the square:
The side of the square is equal to the hypotenuse of the triangle, so the area of the square is given by A' = c^2.
Ratio of the areas:
The ratio of the areas of the triangle and the square is given by A/A':
(A)/(c^2) = (1/2) * a * b / c^2
We know that a + b = (5/2)c, so substituting this value:
(A)/(c^2) = (1/2) * (a + b) * (a - b) / c^2
(A)/(c^2) = (1/2) * (5/2)c * (a - b) / c^2
(A)/(c^2) = (5/4) * (a - b) / c
We also know that a^2 + b^2 = c^2 (Pythagoras theorem), so substituting this value:
(A)/(c^2) = (5/4) * (a - b) / c
(A)/(c^2) = (5/4) * (sqrt(c^2 - b^2) - b) / c
Simplifying further:
(A)/(c^2) = (5/4) * (sqrt(c^2 - b^2)/c - b/c)
(A)/(c^2) = (5/4) * (sqrt(1 - (b/c)^2) - b/c)
Since b/c is the sine of the angle opposite to side b, let's denote it as sin(B):
(A)/(c^2) = (5/4) * (sqrt(1 - sin^2(B)) - sin(B))
Using the trigonometric identity sin^2(B) + cos^2(B) = 1, we can write the ratio as:
(A)/(c^2) = (5/4) * (cos(B
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The difference between the lengths of the inradius and circumradius of a right-angled triangle is equal to a fourth of the length of the hypotenuse of the triangle. A square is constructed using the hypotenuse of the triangle as the base. What is the ratio of the areas of the triangle and the square?a)1:2b)1:3c)7:16d)5:16Correct answer is option 'D'. Can you explain this answer?
Question Description
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