Answer: Choice H) 12 inches
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Step-by-step explanation:
We'll use the pythagorean theorem converse.
Consider a triangle with sides a,b,c where c is the longest side.
There are 3 scenarios possible:
- If a^2+b^2 > c^2, then the triangle is acute.
- If a^2+b^2 = c^2, then it is a right triangle.
- If a^2+b^2 < c^2, then the triangle is obtuse.
The middle or second scenario is probably what you're most familiar with. That's the famous pythagorean theorem without the term "converse" attached to it.
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The two given sides are 8 inches and 10 inches.
Let's say the third side was 5 inches (choice F)
This means we have a triangle with sides a = 5, b = 8, c = 10
Then compute the following:
- a^2+b^2 = (5)^2 + (8)^2 = 89
- c^2 = (10)^2 = 100
Since 89 < 100, we see that a^2+b^2 < c^2 is the case. Therefore the triangle with sides 5,8,10 is obtuse.
This will allow us to rule out choice F since we're looking for an acute triangle.
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Now onto choice G.
This time the sides are: a = 6, b = 8, c = 10
- a^2+b^2 = (6)^2 + (8)^2 = 100
- c^2 = (10)^2 = 100
Both results (a^2+b^2 and c^2) are the same at 100; so we see that a^2+b^2 = c^2.
Therefore we have a right triangle. We can rule out choice G.
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Now onto choice H.
The triangle now has sides: a = 8, b = 10, c = 12
- a^2+b^2 = (8)^2 + (10)^2 = 164
- c^2 = (12)^2 = 144
Since 164 > 144, we see that a^2+b^2 > c^2 is the case. Therefore the triangle with side lengths 8,10,12 is acute.
Choice H is the final answer.
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For the sake of completeness, let's go over choice J.
- a = 8, b = 10, c = 13
- a^2+b^2 = (8)^2 + (10)^2 = 164
- c^2 = (13)^2 = 169
We find 164 < 169 which leads to the scenario a^2+b^2 < c^2. The triangle for choice J is obtuse. We can rule out choice J.
You can use something like GeoGebra to help visually verify these results.