Question

isosceles triangle abc has a perimeter of 96 centimeters. the base of the triangle is \\(\overline{ac}\\) and measures 24 centimeters. what is the measure of \\(\overline{ab}\\)? \\(\bigcirc\\) 12 cm \\(\bigcirc\\) 36 cm \\(\bigcirc\\) 48 cm \\(\bigcirc\\) 72 cm

Explanation:

Step1: Recall perimeter of isosceles triangle

In an isosceles triangle, two sides are equal. Here, base is \( \overline{AC} = 24 \) cm, and let the equal sides (legs) be \( \overline{AB} \) and \( \overline{BC} \), so \( AB = BC \). The perimeter \( P \) of a triangle is the sum of all its sides, so \( P = AB + BC + AC \).

Step2: Substitute known values

Given \( P = 96 \) cm and \( AC = 24 \) cm, and \( AB = BC \), let \( AB = x \). Then the perimeter formula becomes \( 96 = x + x + 24 \).

Step3: Solve for \( x \)

Simplify the equation: \( 96 = 2x + 24 \). Subtract 24 from both sides: \( 96 - 24 = 2x \), so \( 72 = 2x \). Divide both sides by 2: \( x = \frac{72}{2} = 36 \).

Answer:

36 cm (corresponding to the option "36 cm")