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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].

Sagot :

GinnyAnswer
To determine if the triangle with side lengths 2 inches, 5 inches, and 4 inches is an acute triangle, we need to follow a systematic approach:

1. Understand acute triangles:
- A triangle is acute if all its interior angles are less than 90 degrees.
- For a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(c\)[/tex] is the longest side), it is acute if the sum of the squares of the two smaller sides is greater than the square of the longest side, i.e., [tex]\(a^2 + b^2 > c^2\)[/tex].

2. Identify the sides:
- Here, [tex]\(a = 2\)[/tex] inches, [tex]\(b = 4\)[/tex] inches, and [tex]\(c = 5\)[/tex] inches (the longest side).

3. Apply the conditions for an acute triangle:
- We need to verify if the inequality [tex]\(a^2 + b^2 > c^2\)[/tex] holds.

4. Compute the squares:
- Calculate [tex]\(a^2\)[/tex]: [tex]\(2^2 = 4\)[/tex]
- Calculate [tex]\(b^2\)[/tex]: [tex]\(4^2 = 16\)[/tex]
- Calculate [tex]\(c^2\)[/tex]: [tex]\(5^2 = 25\)[/tex]

5. Check the condition:
- Compute [tex]\(a^2 + b^2\)[/tex]: [tex]\(4 + 16 = 20\)[/tex]
- Compare with [tex]\(c^2\)[/tex]: [tex]\(20 < 25\)[/tex]

Since [tex]\(a^2 + b^2\)[/tex] is less than [tex]\(c^2\)[/tex] ([tex]\(20 < 25\)[/tex]), the condition for an acute triangle is not met.

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