Ramon wants to make an acute triangle with three pieces of wood. So far, he has cut wood lengths of 7 inches and 3 inches. He still needs to cut the longest side. What length must the longest side be in order for the triangle to be acute?

A. exactly [tex]\sqrt{58}[/tex] inches
B. greater than [tex]\sqrt{58}[/tex] inches but less than 10 inches
C. less than [tex]\sqrt{58}[/tex] inches but greater than 7 inches
D. not enough information given



Answer :

To determine the length of the longest side so that the triangle is acute, let's recall that for a triangle to be acute, the square of the longest side must be less than the sum of the squares of the other two sides.

Given:
Side [tex]\( a = 7 \)[/tex] inches
Side [tex]\( b = 3 \)[/tex] inches

We need to determine the appropriate length for the longest side [tex]\( c \)[/tex].

Step-by-step solution:

1. Calculate the squares of sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ a^2 = 7^2 = 49 \][/tex]
[tex]\[ b^2 = 3^2 = 9 \][/tex]

2. Sum the squares of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ a^2 + b^2 = 49 + 9 = 58 \][/tex]

3. For the triangle to remain acute, [tex]\( c^2 \)[/tex] must be less than [tex]\( a^2 + b^2 \)[/tex]:
[tex]\[ c^2 < 58 \][/tex]
Therefore, [tex]\( c \)[/tex] must be less than [tex]\( \sqrt{58} \)[/tex].

4. Calculate [tex]\( \sqrt{58} \)[/tex] to find the critical value:
[tex]\[ \sqrt{58} \approx 7.615773105863909 \][/tex]

Thus, for the triangle to remain acute:

- The length of the longest side [tex]\( c \)[/tex] must be less than 10 inches.
- Additionally, [tex]\( c \)[/tex] must be greater than [tex]\( 7 \)[/tex] inches. However, if [tex]\( c \)[/tex] is less than or equal to 7, then [tex]\( c \)[/tex] would not be the longest side as [tex]\( a = 7 \)[/tex] is already this length.
- Therefore, [tex]\( c \)[/tex] must be greater than [tex]\( \sqrt{58} \approx 7.615773105863909 \)[/tex] to ensure the correct formation of an acute triangle with the given constraints.

As a result, the correct length for the longest side [tex]\( c \)[/tex] is:

Greater than [tex]\( \sqrt{58} \)[/tex] inches but less than 10 inches.