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

Answer 1

Let the sides be 21, x, 2x. Since 21 is one of the two shorter sides, the longest side is 2x and we must have 2x > 21 ⇒ x > 10.5. Triangle inequalities give 3x > 21 ⇒ x > 7 and 21 + x > 2x ⇒ x < 21, so 10.5 < x < 21.

For the triangle to be acute, the square of the longest side must be less than the sum of the squares of the other two:
(2x)^2 < 21^2 + x^2 ⇒ 4x^2 < x^2 + 441 ⇒ 3x^2 < 441 ⇒ x^2 < 147 ⇒ x < 7√3 ≈ 12.124.

Combining, x can approach 7√3 from below, so the longest side 2x can approach 14√3 ≈ 24.2487, which rounds to 24.2 cm.

Answer: 24.2 cm.