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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 an acute triangle, we need to analyze the relationship between the squares of its side lengths. For a triangle to be acute, the sum of the squares of any two sides must be greater than the square of the third side for all three combinations. Let's use the side lengths given and test each option carefully:

Given side lengths:
- [tex]\(a = 2\)[/tex] inches
- [tex]\(b = 5\)[/tex] inches
- [tex]\(c = 4\)[/tex] inches

First, calculate the squares of the side lengths:
- [tex]\(a^2 = 2^2 = 4\)[/tex]
- [tex]\(b^2 = 5^2 = 25\)[/tex]
- [tex]\(c^2 = 4^2 = 16\)[/tex]

Now we will check each of the conditions required for the triangle to be acute.

1. [tex]\(a^2 + b^2 > c^2\)[/tex]:
[tex]\[ 2^2 + 5^2 > 4^2 \\ 4 + 25 > 16 \\ 29 > 16 \quad (\text{True}) \][/tex]

2. [tex]\(b^2 + c^2 > a^2\)[/tex]:
[tex]\[ 5^2 + 4^2 > 2^2 \\ 25 + 16 > 4 \\ 41 > 4 \quad (\text{True}) \][/tex]

3. [tex]\(a^2 + c^2 > b^2\)[/tex]:
[tex]\[ 2^2 + 4^2 > 5^2 \\ 4 + 16 > 25 \\ 20 > 25 \quad (\text{False}) \][/tex]

Since the third condition [tex]\(a^2 + c^2 > b^2\)[/tex] is false, the triangle is not acute.

Thus, the correct explanation is:
- The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].

So, the correct answer is:
- The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].

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