the longest side of an acute triangle measures 30 inches. the two remaining sides are congruent, but their…
the longest side of an acute triangle measures 30 inches. the two remaining sides are congruent, but their length is unknown. what is the smallest possible perimeter of the triangle, rounded to the nearest tenth? 41.0 in. 51.2 in. 72.4 in. 81.2 in.
Answer
Explanation:
Step1: Apply the Pythagorean theorem for an acute triangle
For an acute triangle with sides (a), (a), and (c = 30) (where (c) is the longest side), by the Pythagorean theorem for acute triangles (a^{2}+a^{2}>c^{2}). Substitute (c = 30), we get (2a^{2}>30^{2}), so (2a^{2}>900), then (a^{2}>450), and (a>\sqrt{450}\approx21.21).
Step2: Calculate the perimeter
The perimeter (P=2a + c). Since (a>\sqrt{450}\approx21.21), take (a = 21.2) (rounded to one - decimal place). Then (P=2\times21.2+30). [ \begin{align*} P&=42.4 + 30\ &=72.4 \end{align*} ]
Answer:
72.4 in.