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

AThe triangle is acute because $2+4>5$.
BThe triangle is not acute because $2^2<4^2+5^2$.
CThe triangle is not acute because $2^2+4^2<5^2$.
DThe triangle is acute because $2^2+5^2>4^2$.

Question

03-11-2026
Mathematics

Answered

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

AThe triangle is acute because $2+4>5$.
BThe triangle is not acute because $2^2<4^2+5^2$.
CThe triangle is not acute because $2^2+4^2<5^2$.
DThe triangle is acute because $2^2+5^2>4^2$.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T01:05:23-05:00

Verify the triangle exists using the triangle inequality theorem ($2+4>5$).
Identify the longest side ($c=5$) and calculate the squares of all sides.
Compare the sum of the squares of the two shorter sides ($2^2+4^2=20$) to the square of the longest side ($5^2=25$).
Conclude that since $2^2 + 4^2 < 5^2$, the triangle is obtuse and therefore not acute.
Identify the correct statement: $\boxed{\text{The triangle is not acute because } 2^2 + 4^2 < 5^2.}$

Explanation

Verify the triangle inequality.

We are given a triangle with side lengths $2 \text{ in.}$, $5 \text{ in.}$, and $4 \text{ in.}$. First, we verify the triangle inequality to ensure such a triangle exists. The triangle inequality states that for any triangle with sides $a, b, c$, the sum of any two sides must be greater than the third side. Here, we check:
$2 + 4 = 6 > 5$
$2 + 5 = 7 > 4$
$4 + 5 = 9 > 2$
Since all conditions are satisfied, the triangle exists.

Classify the triangle using side lengths.

To classify the triangle by its angles, we identify the longest side, which is $c = 5$. The other two sides are $a = 2$ and $b = 4$. According to the Pythagorean classification theorem:

If $a^2 + b^2 > c^2$, the triangle is acute.
If $a^2 + b^2 = c^2$, the triangle is a right triangle.
If $a^2 + b^2 < c^2$, the triangle is obtuse (and therefore not acute).

We calculate the squares of the sides:
$a^2 = 2^2 = 4$
$b^2 = 4^2 = 16$
$c^2 = 5^2 = 25$

Now, compare the sum of the squares of the two shorter sides to the square of the longest side:
$a^2 + b^2 = 4 + 16 = 20$
$c^2 = 25$

Since $20 < 25$, or equivalently $2^2 + 4^2 < 5^2$, the triangle is obtuse.

Select the correct explanation.

Based on our calculation that $2^2 + 4^2 < 5^2$, we can conclude that the triangle is not acute. Comparing this with the given choices:

The triangle is acute because $2 + 4 > 5$ (Incorrect: this only proves triangle existence).
The triangle is not acute because $2^2 < 4^2 + 5^2$ (Incorrect: this does not compare against the longest side).
The triangle is not acute because $2^2 + 4^2 < 5^2$ (Correct).
The triangle is acute because $2^2 + 5^2 > 4^2$ (Incorrect).

The best explanation is that the triangle is not acute because $2^2 + 4^2 < 5^2$.

Examples

Understanding the classification of triangles is essential for structural engineering and carpentry. For example, if you are building a triangular roof truss, knowing whether your triangle is acute, right, or obtuse helps you determine the types of load-bearing connectors needed. A right triangle allows for specific right-angle joint calculations, while knowing if a design is obtuse helps ensure that the structural frame won't collapse under specific wind or snow loads. By checking if $a^2 + b^2 < c^2$ for a truss with lengths 2, 4, and 5 meters, an engineer can instantly recognize the triangle is obtuse and adjust the support structure accordingly.