Question

equilateral triangle abc has a perimeter of 96 millimeters. a perpendicular bisector is drawn from angle a to side \\(\overline{bc}\\) at point m.
what is the length of \\(\overline{mc}\\)?
\\(\bigcirc\\) 16 mm
\\(\bigcirc\\) 24 mm
\\(\bigcirc\\) 32 mm
\\(\bigcirc\\) 48 mm

Explanation:

Step1: Find side length of equilateral triangle

An equilateral triangle has three equal sides. Let the side length be \( s \). The perimeter \( P = 3s \). Given \( P = 96 \) mm, so \( 3s = 96 \). Solving for \( s \), we get \( s=\frac{96}{3}=32 \) mm. So each side (like \( BC \)) is 32 mm.

Step2: Use perpendicular bisector property

A perpendicular bisector of a side in a triangle (here, from \( A \) to \( BC \) at \( M \)) divides the side into two equal parts. So \( MC=\frac{BC}{2} \). Since \( BC = 32 \) mm, then \( MC=\frac{32}{2}=16 \) mm? Wait, no, wait. Wait, equilateral triangle, perimeter 96, so each side is 96/3 = 32. Then the perpendicular bisector from A to BC: in an equilateral triangle, the perpendicular from a vertex to the opposite side bisects the opposite side. So \( BM = MC \), so \( MC=\frac{BC}{2}=\frac{32}{2}=16 \)? Wait, no, wait, 96 divided by 3 is 32, so BC is 32. Then bisected, so MC is 16? But wait, the options have 16, 24, 32, 48. Wait, maybe I made a mistake. Wait, no, let's recheck. Perimeter of equilateral triangle: 3*side = 96, so side = 32. Then the perpendicular bisector from A to BC: in an equilateral triangle, the altitude, median, angle bisector, perpendicular bisector all coincide. So M is the midpoint of BC, so MC = BC/2 = 32/2 = 16 mm. Wait, but the options have 16 as the first option. Wait, but maybe I messed up. Wait, no, 96 divided by 3 is 32, so each side is 32. Then midpoint, so 32/2 is 16. So the length of MC is 16 mm? Wait, but let me check again. Wait, maybe the problem is that I thought the side is 32, but maybe I miscalculated. 96 divided by 3 is 32, correct. Then midpoint, so 32/2 is 16. So the answer should be 16 mm? Wait, but the options are 16, 24, 32, 48. Wait, maybe I made a mistake. Wait, no, let's see: equilateral triangle, perimeter 96, so each side is 32. Perpendicular bisector from A to BC: so M is the midpoint, so MC is half of BC, which is 16. So the correct answer is 16 mm. Wait, but let me check again. Wait, maybe the problem is that I thought the side is 32, but maybe the perimeter is 96, so each side is 32, so BC is 32, so MC is 16. Yes, that makes sense.

Wait, but wait, maybe I made a mistake in the step. Let's redo:

Step 1: Perimeter of equilateral triangle \( ABC \) is 96 mm. Since all sides are equal, let each side be \( x \). Then \( 3x = 96 \), so \( x = 96/3 = 32 \) mm. So \( BC = 32 \) mm.

Step 2: A perpendicular bisector from \( A \) to \( BC \) at \( M \) means \( M \) is the midpoint of \( BC \) (because in a triangle, a perpendicular bisector of a side passes through its midpoint). Therefore, \( MC = \frac{BC}{2} = \frac{32}{2} = 16 \) mm.

Yes, that's correct. So the length of \( \overline{MC} \) is 16 mm.

Answer:

16 mm (corresponding to the first option: 16 mm)