Question

Which best explains whether a triangle with side lengths 2 in., 5 in., and 4 in. is an acute triangle? The triangle is acute because 22 + 52 > 42. The triangle is acute because 2 + 4 > 5. The triangle is not acute because 22 + 42 < 52. The triangle is not acute because 22 < 42 + 52.

Answer 1

Answer: The correct option is

(C) The triangle is not acute because 2² + 4² < 5².

Step-by-step explanation: We are to select the statement that best explains the type of the triangle having lengths of three sides as 2 inch, 5 inch and 4 inch.

We know that a triangle with side lengths a, b and c (c > a, b)is

(i) an acute-angled if a² + b² > c², and

(ii) an obtuse-angled if a² + b² < c².

For the given triangle,

a = 2 inch, b = 4 inch and c = 5 inch.

So, we have

`a^2+b^2=2^2+4^2=4+16=20,\\\\c^2=5^2=25.`

Since,

`20<25\\\\\Rightarrow a^2+b^2<c^2,`

so the given triangle is not acute, but obtuse.

Thus, the triangle is not acute because 2² + 4² < 5².

Option (C) is correct.

Answer 2

Let

`a=2\ in\\b=4\ in\\c=5\ in`

we know that

If
`c^(2) =a^(2)+b^(2)` -----> is a right triangle

If
`c^(2) > a^(2)+b^(2)` -----> is an obtuse triangle

If
`c^(2) < a^(2)+b^(2)` -----> is an acute triangle

so

substitute the values

`5^(2) > 2^(2)+4^(2)` ------> is an obtuse triangle

therefore

the answer is

The triangle is not acute because
`2^(2)+4^(2)< 5^(2)`