an acute triangle has side lengths 21 cm, x cm, and 2x cm. if 21 is one of the shorter sides of the…

Explanation:

Step1: Apply the acute - triangle inequality

For an acute triangle with side lengths (a), (b), and (c) ((c) being the longest side), (a^{2}+b^{2}>c^{2}). Here, since 21 is one of the shorter sides and the sides are 21, (x), and (2x) ((2x) is the longest side), we have (21^{2}+x^{2}>(2x)^{2}). [21^{2}+x^{2}>4x^{2}] [441 + x^{2}>4x^{2}] [441>3x^{2}] [x^{2}<147] [x < \sqrt{147}\approx12.124] Also, considering the triangle - inequality theorem: (a + b>c), so (21 + x>2x), which gives (x<21). The more restrictive condition is (x < \sqrt{147}).

Step2: Find the length of the longest side

The longest side is (2x). Substituting the maximum value of (x\approx12.124) into (2x), we get (2x\approx2\times12.124 = 24.248\approx24.2) cm.

Answer:

24.2 cm