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The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c?
Verified Answer
The length of the hypotenuse of a right angle triangle is 240 units . ...
A^2 + b^2 = c^2 = 240^2 = 57600
a + b + 240 > 550
a + b > 310
a > 310 - b
b > 310 - a
first perfect square after 550 is 24^2
which is 576 set a + b + 240 equal to 576
a + b = 336 (576 - 240 = 336)
a = 336 - b
b = 336 - a (use this to subsitute)
a^2 + (336 - a)^2 = 57600
a^2 + 112896 - 672a + a^2 = 57600
(336^2 = 112896, 2*336 = 672)
2a^2 - 672a + 55296 = 0
a^2 - 336a + 27648 = 0
27648 = 2 * 13824
13824 = 2 * 6912
6912 = 2 * 3456
3456 = 2 * 1728 1728 = 2 * 864
864 = 2 * 432 432 = 2 * 216
216 = 2 * 108 108 = 2 * 54
54 = 2 * 27
27 = 3 * 3 * 3
27648 = 2^10 * 3^3 = 1024 * 27
144 = 2^4 * 3^2
27648 = 2^4 * 3^2 * 2^6 * 3 = 16 * 9 * 64 * 3 = 144 * 192
144 + 192 = 336 (a - 192)(a - 144) = 0
a = 144 or a = 192
now b = 336 - a
so b = 192 or b = 144
side a needs to be a minimum of 144, and side b needs to be a minimum of 192
a + b + 240 > 550
a + b > 310
a > 310 - b
b > 310 - a
first perfect square after 550 is 24^2
which is 576 set a + b + 240 equal to 576
a + b = 336 (576 - 240 = 336)
a = 336 - b
b = 336 - a (use this to subsitute)
a^2 + (336 - a)^2 = 57600
a^2 + 112896 - 672a + a^2 = 57600
(336^2 = 112896, 2*336 = 672)
2a^2 - 672a + 55296 = 0
a^2 - 336a + 27648 = 0
27648 = 2 * 13824
13824 = 2 * 6912
6912 = 2 * 3456
3456 = 2 * 1728 1728 = 2 * 864
864 = 2 * 432 432 = 2 * 216
216 = 2 * 108 108 = 2 * 54
54 = 2 * 27
27 = 3 * 3 * 3
27648 = 2^10 * 3^3 = 1024 * 27
144 = 2^4 * 3^2
27648 = 2^4 * 3^2 * 2^6 * 3 = 16 * 9 * 64 * 3 = 144 * 192
144 + 192 = 336 (a - 192)(a - 144) = 0
a = 144 or a = 192
now b = 336 - a
so b = 192 or b = 144
side a needs to be a minimum of 144, and side b needs to be a minimum of 192
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Most Upvoted Answer
The length of the hypotenuse of a right angle triangle is 240 units . ...
Solution:
Given:
- Length of the hypotenuse of a right angle triangle = 240 units
- Perimeter of the triangle is a perfect square
- Perimeter of the triangle > 550 units
To find:
- Length of a side of the right-angled triangle
Analysis:
Let's assume the sides of the right-angled triangle are a, b, and c, where c is the hypotenuse.
According to the Pythagorean theorem, the sum of the squares of the two sides of a right-angled triangle is equal to the square of the hypotenuse.
So, we have:
a^2 + b^2 = c^2 ...(1)
We are given the length of the hypotenuse, which is 240 units. Therefore, we have:
a^2 + b^2 = 240^2 ...(2)
Also, the perimeter of the triangle is the sum of the lengths of its sides:
Perimeter = a + b + c
Since the perimeter is a perfect square, we can write:
Perimeter = k^2 ...(3), where k is a positive integer
Now, we need to find the possible lengths of the sides of the triangle.
Solution:
Let's consider the given options one by one:
a. 192 units:
If one side of the triangle is 192 units, then using equation (2), we have:
192^2 + b^2 = 240^2
Simplifying the equation, we get:
b^2 = 240^2 - 192^2
b^2 = (240 + 192)(240 - 192)
b^2 = 432 * 48
Now, we need to check if the perimeter is greater than 550 units:
Perimeter = a + b + c
Perimeter = 192 + b + 240
Perimeter = 432 + b
Since b is a positive integer, the minimum value of b can be 1.
So, the minimum possible perimeter = 432 + 1 = 433
Since 433 is not a perfect square and it is less than 550, the length of a side cannot be 192 units.
b. 168 units:
If one side of the triangle is 168 units, then using equation (2), we have:
168^2 + b^2 = 240^2
b^2 = 240^2 - 168^2
b^2 = (240 + 168)(240 - 168)
b^2 = 408 * 72
Now, we need to check if the perimeter is greater than 550 units:
Perimeter = a + b + c
Perimeter = 168 + b + 240
Perimeter = 408 + b
Since b is a positive integer, the minimum value of b can be 1.
So, the minimum possible perimeter = 408 + 1 = 409
Since 409 is not a perfect square and it is less than 550, the length of a side cannot be 168 units.
c. 144 units:
If one side of the triangle is 144 units, then using equation (2), we have:
144^2 + b^2 = 240^2
b^2 = 240^2 - 144^
Given:
- Length of the hypotenuse of a right angle triangle = 240 units
- Perimeter of the triangle is a perfect square
- Perimeter of the triangle > 550 units
To find:
- Length of a side of the right-angled triangle
Analysis:
Let's assume the sides of the right-angled triangle are a, b, and c, where c is the hypotenuse.
According to the Pythagorean theorem, the sum of the squares of the two sides of a right-angled triangle is equal to the square of the hypotenuse.
So, we have:
a^2 + b^2 = c^2 ...(1)
We are given the length of the hypotenuse, which is 240 units. Therefore, we have:
a^2 + b^2 = 240^2 ...(2)
Also, the perimeter of the triangle is the sum of the lengths of its sides:
Perimeter = a + b + c
Since the perimeter is a perfect square, we can write:
Perimeter = k^2 ...(3), where k is a positive integer
Now, we need to find the possible lengths of the sides of the triangle.
Solution:
Let's consider the given options one by one:
a. 192 units:
If one side of the triangle is 192 units, then using equation (2), we have:
192^2 + b^2 = 240^2
Simplifying the equation, we get:
b^2 = 240^2 - 192^2
b^2 = (240 + 192)(240 - 192)
b^2 = 432 * 48
Now, we need to check if the perimeter is greater than 550 units:
Perimeter = a + b + c
Perimeter = 192 + b + 240
Perimeter = 432 + b
Since b is a positive integer, the minimum value of b can be 1.
So, the minimum possible perimeter = 432 + 1 = 433
Since 433 is not a perfect square and it is less than 550, the length of a side cannot be 192 units.
b. 168 units:
If one side of the triangle is 168 units, then using equation (2), we have:
168^2 + b^2 = 240^2
b^2 = 240^2 - 168^2
b^2 = (240 + 168)(240 - 168)
b^2 = 408 * 72
Now, we need to check if the perimeter is greater than 550 units:
Perimeter = a + b + c
Perimeter = 168 + b + 240
Perimeter = 408 + b
Since b is a positive integer, the minimum value of b can be 1.
So, the minimum possible perimeter = 408 + 1 = 409
Since 409 is not a perfect square and it is less than 550, the length of a side cannot be 168 units.
c. 144 units:
If one side of the triangle is 144 units, then using equation (2), we have:
144^2 + b^2 = 240^2
b^2 = 240^2 - 144^
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The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c?
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The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c?.
The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c?.
Solutions for The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? in English & in Hindi are available as part of our courses for CAT.
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The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c?, a detailed solution for The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? has been provided alongside types of The length of the hypotenuse of a right angle triangle is 240 units . The perimeter of the given triangle is a perfect square, if the perimeter of the given triangle is greater than 550 units then which of the following can be the length of a side of the given right angled triangle? a. 192 units b. 168 units. c. 144units. d. Both a and c? theory, EduRev gives you an
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