Question

an acute triangle has side lengths 21 cm, x cm, and 2x cm. if 21 is one of the shorter sides of the triangle, what is the greatest possible length of the longest side, rounded to the nearest tenth?
18.8 cm
24.2 cm
42.0 cm
72.7 cm

Explanation:

Step1: Identify longest side

Longest side is \(2x\) (since \(2x > x\) and 21 is shorter).

Step2: Acute triangle condition

For acute triangle: \((2x)^2 < 21^2 + x^2\)

Step3: Solve inequality

\(4x^2 < 441 + x^2 \Rightarrow 3x^2 < 441 \Rightarrow x^2 < 147 \Rightarrow x < \sqrt{147} \approx 12.124\)

Step4: Longest side length

Longest side \(2x \approx 2 \times 12.124 \approx 24.2\)

Answer:

B. 24.2 cm