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If one side of a right-angled triangle (with all sides integers) is 15 cm, then what is the maximum perimeter of the triangle?
- a)240 cm
- b)225 cm
- c)113 cm
- d)112 cm
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If one side of a right-angled triangle (with all sides integers) is 15...
To find the maximum perimeter of a right-angled triangle with one side measuring 15 cm, we need to determine the lengths of the other two sides.
Let's assume the other two sides are a and b, with a being the hypotenuse (the side opposite the right angle).
Using the Pythagorean theorem, we can write the equation:
a^2 = 15^2 + b^2
Simplifying this equation, we have:
a^2 = 225 + b^2
Since all sides of the triangle are integers, a and b must also be integers.
Now let's consider the possible values of b. Since a^2 = 225 + b^2, a^2 must be a perfect square.
The factors of 225 are 1, 3, 5, 9, 15, 25, 45, 75, and 225. We can see that 225 is divisible by 15, which means a = 15 and b = 0.
Therefore, the maximum perimeter of the triangle is when a = 15, b = 0, and the hypotenuse is the longest side. In this case, the perimeter is:
P = 15 + 15 + 0 = 30 cm
However, the question asks for the maximum perimeter, not the maximum sum of the sides. Since the other options do not have a perimeter of 30 cm, we need to find another solution.
If we consider a = 15 and b = 8, we get:
a^2 = 225 + 8^2
a^2 = 225 + 64
a^2 = 289
a = 17
In this case, the perimeter of the triangle is:
P = 15 + 8 + 17 = 40 cm
Since 40 cm is greater than 30 cm, we can conclude that the maximum perimeter of the triangle is 40 cm, which is not listed among the answer options.
Therefore, none of the given options is correct.
Let's assume the other two sides are a and b, with a being the hypotenuse (the side opposite the right angle).
Using the Pythagorean theorem, we can write the equation:
a^2 = 15^2 + b^2
Simplifying this equation, we have:
a^2 = 225 + b^2
Since all sides of the triangle are integers, a and b must also be integers.
Now let's consider the possible values of b. Since a^2 = 225 + b^2, a^2 must be a perfect square.
The factors of 225 are 1, 3, 5, 9, 15, 25, 45, 75, and 225. We can see that 225 is divisible by 15, which means a = 15 and b = 0.
Therefore, the maximum perimeter of the triangle is when a = 15, b = 0, and the hypotenuse is the longest side. In this case, the perimeter is:
P = 15 + 15 + 0 = 30 cm
However, the question asks for the maximum perimeter, not the maximum sum of the sides. Since the other options do not have a perimeter of 30 cm, we need to find another solution.
If we consider a = 15 and b = 8, we get:
a^2 = 225 + 8^2
a^2 = 225 + 64
a^2 = 289
a = 17
In this case, the perimeter of the triangle is:
P = 15 + 8 + 17 = 40 cm
Since 40 cm is greater than 30 cm, we can conclude that the maximum perimeter of the triangle is 40 cm, which is not listed among the answer options.
Therefore, none of the given options is correct.
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Community Answer
If one side of a right-angled triangle (with all sides integers) is 15...
By using options,
Check option 1:
Perimeter of the triangle = (113 + 112 + 15) cm = 240 cm
Check option 1:
Perimeter of the triangle = (113 + 112 + 15) cm = 240 cm
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If one side of a right-angled triangle (with all sides integers) is 15 cm, then what is the maximum perimeter of the triangle?a)240 cmb)225 cmc)113 cmd)112 cmCorrect answer is option 'A'. Can you explain this answer?
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