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Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is
Correct answer is '9'. Can you explain this answer?
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Verified Answer
Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and...
In any right triangle, the circumradius is half of the hypotenuse. Here,L=
the length of the hypotenuse = 
Hence, the integer close to L = 9
Most Upvoted Answer
Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and...
Given:
- The equation of the straight line is 3x + 5y - 45 = 0.
- The triangle T is formed by this line and the coordinate axes.
- The radius of the circumcircle of T is denoted by L.
To find:
The integer closest to the length of L.
Solution:
Step 1: Find the vertices of the triangle T:
To find the vertices of the triangle T, we need to find the points where the line intersects the coordinate axes.
Intersecting the x-axis:
When y = 0, we can solve the equation 3x + 5y - 45 = 0 to find the x-coordinate of the point of intersection with the x-axis.
3x + 5(0) - 45 = 0
3x - 45 = 0
3x = 45
x = 15
So, the point of intersection with the x-axis is (15, 0).
Intersecting the y-axis:
When x = 0, we can solve the equation 3x + 5y - 45 = 0 to find the y-coordinate of the point of intersection with the y-axis.
3(0) + 5y - 45 = 0
5y - 45 = 0
5y = 45
y = 9
So, the point of intersection with the y-axis is (0, 9).
Step 2: Find the distance between the vertices:
Using the distance formula, we can find the distance between the vertices of the triangle T.
Distance between (15, 0) and (0, 9):
d = √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - 15)^2 + (9 - 0)^2)
= √(225 + 81)
= √306
So, the distance between the vertices (15, 0) and (0, 9) is √306.
Step 3: Find the radius of the circumcircle:
The radius of the circumcircle of a triangle is given by the formula:
R = (abc) / (4A)
Where a, b, and c are the lengths of the sides of the triangle, and A is the area of the triangle.
In this case, we have a right-angled triangle, so we can use the formula for the area of a right-angled triangle:
A = (1/2) * base * height
The base and height of the triangle T are the distances between the vertices.
Base: The base of the triangle is the distance between (15, 0) and (0, 0), which is 15.
Height: The height of the triangle is the distance between (0, 0) and (0, 9), which is 9.
Area: A = (1/2) * 15 * 9 = 67.5
Radius: R = (15 * √306 * 9) / (4 * 67.5) = (√
- The equation of the straight line is 3x + 5y - 45 = 0.
- The triangle T is formed by this line and the coordinate axes.
- The radius of the circumcircle of T is denoted by L.
To find:
The integer closest to the length of L.
Solution:
Step 1: Find the vertices of the triangle T:
To find the vertices of the triangle T, we need to find the points where the line intersects the coordinate axes.
Intersecting the x-axis:
When y = 0, we can solve the equation 3x + 5y - 45 = 0 to find the x-coordinate of the point of intersection with the x-axis.
3x + 5(0) - 45 = 0
3x - 45 = 0
3x = 45
x = 15
So, the point of intersection with the x-axis is (15, 0).
Intersecting the y-axis:
When x = 0, we can solve the equation 3x + 5y - 45 = 0 to find the y-coordinate of the point of intersection with the y-axis.
3(0) + 5y - 45 = 0
5y - 45 = 0
5y = 45
y = 9
So, the point of intersection with the y-axis is (0, 9).
Step 2: Find the distance between the vertices:
Using the distance formula, we can find the distance between the vertices of the triangle T.
Distance between (15, 0) and (0, 9):
d = √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - 15)^2 + (9 - 0)^2)
= √(225 + 81)
= √306
So, the distance between the vertices (15, 0) and (0, 9) is √306.
Step 3: Find the radius of the circumcircle:
The radius of the circumcircle of a triangle is given by the formula:
R = (abc) / (4A)
Where a, b, and c are the lengths of the sides of the triangle, and A is the area of the triangle.
In this case, we have a right-angled triangle, so we can use the formula for the area of a right-angled triangle:
A = (1/2) * base * height
The base and height of the triangle T are the distances between the vertices.
Base: The base of the triangle is the distance between (15, 0) and (0, 0), which is 15.
Height: The height of the triangle is the distance between (0, 0) and (0, 9), which is 9.
Area: A = (1/2) * 15 * 9 = 67.5
Radius: R = (15 * √306 * 9) / (4 * 67.5) = (√
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Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer?
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Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. 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 Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer?.
Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. 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 Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer?.
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Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer?, a detailed solution for Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer? has been provided alongside types of Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L isCorrect answer is '9'. Can you explain this answer? theory, EduRev gives you an
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