Question

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

Explanation:

Step1: Find side length of equilateral triangle

In an equilateral triangle, all sides are equal. The perimeter \( P \) of a triangle is the sum of its three sides. For equilateral triangle \( ABC \), let the length of each side be \( s \). So, \( P = 3s \). Given \( P = 96 \) mm, we solve for \( s \):

$$ 3s = 96 \implies s=\frac{96}{3}=32 \text{ mm} $$

So, each side (e.g., \( BC \)) is 32 mm.

Step2: Use property of perpendicular bisector

A perpendicular bisector of a side in a triangle (especially in an equilateral triangle) divides the side into two equal parts. So, the perpendicular bisector from \( A \) to \( BC \) at \( M \) means \( M \) is the midpoint of \( BC \). Therefore, \( MC=\frac{BC}{2} \).

Since \( BC = 32 \) mm, then:

$$ MC=\frac{32}{2}=16 \text{ mm} $$

Wait, no, wait. Wait, in an equilateral triangle, the perimeter is 96, so each side is \( 96\div3 = 32 \) mm. Then the perpendicular bisector from \( A \) to \( BC \) (since in an equilateral triangle, the altitude, median, perpendicular bisector, angle bisector all coincide) so \( M \) is the midpoint of \( BC \), so \( BM = MC=\frac{BC}{2}=\frac{32}{2}=16 \) mm? Wait, but wait, maybe I made a mistake. Wait, no, wait, 96 divided by 3 is 32, so each side is 32. Then the midpoint divides it into two 16s. So \( MC = 16 \) mm? But wait, the options have 16, 24, 32, 48. Wait, but let me check again. Wait, perimeter is 96, so each side is 96/3 = 32. Then the perpendicular bisector from A to BC: in a triangle, the perpendicular bisector of a side passes through the midpoint of the side. So M is the midpoint of BC, so BM = MC = 32/2 = 16 mm. So the answer is 16 mm? Wait, but wait, maybe I messed up. Wait, no, 32 divided by 2 is 16. So the length of MC is 16 mm.

Wait, but wait, maybe I made a mistake in the side length. Wait, perimeter is 96, three equal sides, so each side is 96/3 = 32. Then the perpendicular bisector of BC (since it's from A to BC) will split BC into two equal parts, so each part is 32/2 = 16. So MC is 16 mm. So the correct option is 16 mm.

Answer:

16 mm (corresponding to the option "16 mm")