Answers:
A triangle is possible.
The triangle is <u>acute</u> and <u>scalene</u>
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Explanation:
We use the triangle inequality theorem to see if a triangle is possible with these side lengths.
Consider the side lengths a, b, and c. A triangle is possible if and only if the following 3 conditions hold true
- a+b > c
- a+c > b
- b+c > a
Basically pick any two sides. If the sum is larger than the third side, then a triangle is possible.
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In this case we have: a = 8, b = 14, c = 16. The order of the a,b,c values doesn't really matter.
Then:
- a+b = 8+14 = 22 which is larger than c = 16. So a+b > c is true.
- a+c = 8+16 = 24 is larger than b = 14. So a+c > b is true.
- b+c = 14+16 = 30 is larger than a = 8. So b+c > a is true.
All three inequalities mentioned are true; therefore, <u>a triangle is possible</u> with these side lengths.
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We'll use those a,b,c values in the converse of the pythagorean theorem to determine what kind of triangle this is. So far we know it's <u>scalene</u> because all three side lengths are different. It would be isosceles if it had exactly two equal sides, and equilateral if all 3 sides were the same length.
The last equation is false, which tells us this triangle is not a right triangle. It's either acute or obtuse.
Since is the case here, we can conclude the triangle is <u>acute</u>
If was the case, then the triangle would be obtuse.
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Summary:
- A <u>triangle is possible</u> with side lengths a = 8, b = 14, c = 16.
- This triangle is <u>scalene</u> and <u>acute</u>