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The length of the hypotenuse of a right-angled 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?
(2014)
- a)192 units
- b)168 units
- c)144 units
- d)Both (a) and (c)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The length of the hypotenuse of a right-angled triangle is 240 units. ...
Let the length of the other two sides of the triangle be 'x' and 'y' respectively. then,
x2 + y2 = (240)2 (hypotenuse of triangle = 240)
Also, x + y + 240 must be perfect square (because perimeter of triangle is perfect square).
When the two sides other than hypotenuse are equal length than perimeter of right angled triangle is maximized.
Therefore,
The maximum perimeter of right angled triangle
Also,
The perimeter of triangle should be greater than twice the length of hypotenuse of triangle.
Then, the perimeter of triangle should be greater than 480 and less than 579. according to question.
Perimeter of triangle should be 484, 529 and 576
(because perimeter is perfect square)
Again, according to question, Perimeter of triangle is greater than 550, So perimeter = 576.
Here, perimeter of triangle = 576.
According to pythogorean triples.
The value of x and y will be 192 and 144 units.
x2 + y2 = (240)2 (hypotenuse of triangle = 240)
Also, x + y + 240 must be perfect square (because perimeter of triangle is perfect square).
When the two sides other than hypotenuse are equal length than perimeter of right angled triangle is maximized.
Therefore,
The maximum perimeter of right angled triangle
Also,
The perimeter of triangle should be greater than twice the length of hypotenuse of triangle.
Then, the perimeter of triangle should be greater than 480 and less than 579. according to question.
Perimeter of triangle should be 484, 529 and 576
(because perimeter is perfect square)
Again, according to question, Perimeter of triangle is greater than 550, So perimeter = 576.
Here, perimeter of triangle = 576.
According to pythogorean triples.
The value of x and y will be 192 and 144 units.
Most Upvoted Answer
The length of the hypotenuse of a right-angled triangle is 240 units. ...
Given information:
- The length of the hypotenuse of a right-angled triangle is 240 units.
- The perimeter of the triangle is a perfect square.
- The perimeter of the triangle is greater than 550 units.
To find:
The possible lengths of the sides of the triangle.
Solution:
Let's consider the sides of the right-angled triangle to be a, b, and c, with c being the hypotenuse.
Using Pythagoras theorem, we know that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
So, for our given triangle, we have:
a^2 + b^2 = 240^2
Now, let's find the perimeter of the triangle. The perimeter is the sum of the lengths of all three sides:
Perimeter = a + b + c
Since the perimeter is a perfect square, let's assume it is equal to k^2, where k is an integer.
From the given information, we know that the perimeter is greater than 550 units. So, we have:
a + b + c > 550
a + b > 550 - c
We also know that a^2 + b^2 = 240^2. Rearranging this equation, we get:
a^2 = 240^2 - b^2
From the above two equations, we can conclude that:
a^2 > 240^2 - b^2
a^2 > (240 + b)(240 - b)
Now, let's analyze the answer choices:
a) 192 units
Plugging this value into the equation a^2 + b^2 = 240^2, we get:
192^2 + b^2 = 240^2
b^2 = 240^2 - 192^2
b^2 = 57600 - 36864
b^2 = 20736
b = 144
Checking the perimeter: a + b + c = 192 + 144 + 240 = 576, which is not a perfect square.
c) 144 units
Plugging this value into the equation a^2 + b^2 = 240^2, we get:
a^2 + 144^2 = 240^2
a^2 = 240^2 - 144^2
a^2 = 57600
a = 240
Checking the perimeter: a + b + c = 240 + 144 + 240 = 624, which is not a perfect square.
Therefore, neither option a) nor c) can be the length of a side of the given right-angled triangle.
Hence, the correct answer is option d) Both (a) and (c).
- The length of the hypotenuse of a right-angled triangle is 240 units.
- The perimeter of the triangle is a perfect square.
- The perimeter of the triangle is greater than 550 units.
To find:
The possible lengths of the sides of the triangle.
Solution:
Let's consider the sides of the right-angled triangle to be a, b, and c, with c being the hypotenuse.
Using Pythagoras theorem, we know that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
So, for our given triangle, we have:
a^2 + b^2 = 240^2
Now, let's find the perimeter of the triangle. The perimeter is the sum of the lengths of all three sides:
Perimeter = a + b + c
Since the perimeter is a perfect square, let's assume it is equal to k^2, where k is an integer.
From the given information, we know that the perimeter is greater than 550 units. So, we have:
a + b + c > 550
a + b > 550 - c
We also know that a^2 + b^2 = 240^2. Rearranging this equation, we get:
a^2 = 240^2 - b^2
From the above two equations, we can conclude that:
a^2 > 240^2 - b^2
a^2 > (240 + b)(240 - b)
Now, let's analyze the answer choices:
a) 192 units
Plugging this value into the equation a^2 + b^2 = 240^2, we get:
192^2 + b^2 = 240^2
b^2 = 240^2 - 192^2
b^2 = 57600 - 36864
b^2 = 20736
b = 144
Checking the perimeter: a + b + c = 192 + 144 + 240 = 576, which is not a perfect square.
c) 144 units
Plugging this value into the equation a^2 + b^2 = 240^2, we get:
a^2 + 144^2 = 240^2
a^2 = 240^2 - 144^2
a^2 = 57600
a = 240
Checking the perimeter: a + b + c = 240 + 144 + 240 = 624, which is not a perfect square.
Therefore, neither option a) nor c) can be the length of a side of the given right-angled triangle.
Hence, the correct answer is option d) Both (a) and (c).
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The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. Can you explain this answer?
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The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. 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 The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. Can you explain this answer? 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-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. Can you explain this answer?.
The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. 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 The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. Can you explain this answer? 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-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If theperimeter 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-angledtriangle?(2014)a)192 unitsb)168 unitsc)144 unitsd)Both (a) and (c)Correct answer is option 'D'. Can you explain this answer?.
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