Answer: Choice D) 7.5 inches, 18 inches, 19.5 inches
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Explanation:
The Pythagorean theorem converse can be used to determine if we have a right triangle or not.
The converse says that if a^2 + b^2 = c^2 is a true equation, then we have a right triangle.
The value of c is always the longest side. The order of 'a' and b doesn't matter (usually b is larger than or equal to 'a'). So we could have [tex]a \le b < c[/tex] or [tex]b \le a < c[/tex], but convention usually leans toward the first inequality mentioned.
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In choice A, we have the following steps
a^2 + b^2 = c^2
(7.5)^2 + (18)^2 = (21.5)^2
380.25 = 462.25
Since the third equation is false, this means the first equation is false for those a,b,c values. Choice A is not a right triangle. We can cross it off the list.
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In choice B, we have a = 7.5, b = 19.5 and c = 21.5
So,
a^2 + b^2 = c^2
(7.5)^2 + (19.5)^2 = (21.5)^2
436.5 = 462.25
which is also false. Cross choice B off the list.
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With choice C we have: a = 18, b = 19.5, c = 21.5
a^2 + b^2 = c^2
(18)^2 + (19.5)^2 = (21.5)^2
704.25 = 462.25
Cross choice C off the list since the third equation is false.
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In choice D, we have the sides a = 7.5, b = 18, c = 19.5
a^2 + b^2 = c^2
(7.5)^2 + (18)^2 = (19.5)^2
380.25 = 380.25
We get the same number on both sides, so the third equation is true. Consequently, this leads to the first equation being true for those a,b,c values. Choice D is a right triangle, and it's the final answer.
