Question

the longest side of an acute isosceles triangle is 8 centimeters. rounded to the nearest tenth, what is the smallest possible length of one of the two congruent sides?
○ 4.0 cm
○ 4.1 cm
○ 5.6 cm
○ 5.7 cm

Explanation:

Step1: Recall triangle and acute angle conditions

For an isosceles triangle with congruent sides \( x \) and base \( 8 \) (longest side), by triangle inequality: \( x + x>8\Rightarrow 2x > 8\Rightarrow x > 4 \). For it to be acute, the square of the longest side must be less than the sum of the squares of the other two sides. So \( 8^{2}32\Rightarrow x > \sqrt{32}\approx5.656 \).

Step2: Analyze options

• \( 4.0 \): Less than \( 5.656 \), invalid.
• \( 4.1 \): Less than \( 5.656 \), invalid.
• \( 5.6 \): Less than \( 5.656 \), invalid.
• \( 5.7 \): Greater than \( 5.656 \), valid.

Answer:

5.7 cm