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Which best explains whether a triangle with side lengths 2 in., 5 in., and 4 in. is an acute triangle?

A. The triangle is acute because [tex]\(2^2 + 5^2 \ \textgreater \ 4^2\)[/tex].
B. The triangle is acute because [tex]\(2 + 4 \ \textgreater \ 5\)[/tex].
C. The triangle is not acute because [tex]\(2^2 + 4^2 \ \textless \ 5^2\)[/tex].
D. The triangle is not acute because [tex]\(2^2 \ \textless \ 4^2 + 5^2\)[/tex].



Answer :

GinnyAnswer
To determine whether a triangle with side lengths 2 inches, 5 inches, and 4 inches is acute, we need to apply the properties of an acute triangle.

An acute triangle is defined as a triangle where all three interior angles are less than 90 degrees. One key property of acute triangles is that for each pair of sides, the sum of the squares of the shorter sides must be greater than the square of the longest side.

Let's denote the side lengths as:
- [tex]\(a = 2\)[/tex] inches
- [tex]\(b = 5\)[/tex] inches
- [tex]\(c = 4\)[/tex] inches

First, let's identify the longest side, which is [tex]\(b = 5\)[/tex] inches.

Next, we check if the square of the longest side is less than the sum of the squares of the other two sides:
- Calculate [tex]\(a^2\)[/tex]: [tex]\(2^2 = 4\)[/tex]
- Calculate [tex]\(c^2\)[/tex]: [tex]\(4^2 = 16\)[/tex]
- Calculate [tex]\(b^2\)[/tex]: [tex]\(5^2 = 25\)[/tex]

Now compare:
- [tex]\(a^2 + c^2 = 4 + 16 = 20\)[/tex]
- [tex]\(b^2 = 25\)[/tex]

Since [tex]\(a^2 + c^2 < b^2 \)[/tex] (i.e., [tex]\( 20 < 25\)[/tex]), the condition for the triangle to be acute is not satisfied.

Therefore, the triangle is not an acute triangle. The appropriate explanation is:

"The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex]."

This means that the correct option is:
The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].

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