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.
