Question

triangle nr m is created when an equilateral triangle is folded in half. what is the value of y? 4√3 units 8 units 2√3 units 4 units

Explanation:

Step1: Identify Triangle Type

Triangle NMR is formed by folding an equilateral triangle, so it's a 30-60-90 triangle. In such a triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\) (shorter leg : longer leg : hypotenuse). The hypotenuse here is \(NM = 6 + 2 = 8\)? Wait, no—wait, the segment \(NS = 6\), \(SM = 2\), so hypotenuse \(NM = 6 + 2 = 8\)? Wait, no, actually, in a 30-60-90 triangle, the shorter leg is opposite 30°, longer leg opposite 60°, hypotenuse twice the shorter leg. Wait, the right angle is at R, so RM is shorter leg? Wait, no, let's re-examine. The triangle is formed by folding an equilateral triangle, so angle at M is 60°, angle at N is 30°? Wait, no, when you fold an equilateral triangle (all angles 60°) in half, you get a 30-60-90 triangle. The hypotenuse of the 30-60-90 triangle is equal to the side of the equilateral triangle. Wait, the length from N to M: NS is 6, SM is 2, so NM is 8? Wait, no, maybe the original equilateral triangle has side length equal to NM? Wait, no, the right triangle NMR: angle at R is right angle, angle at M is 60° (since it's folded from equilateral), so angle at N is 30°. In a 30-60-90 triangle, the side opposite 30° is the shorter leg, opposite 60° is longer leg, hypotenuse is twice the shorter leg. Wait, the side RM: let's see, the segment SM is 2, and in the equilateral triangle, when folded, the length from M to the midpoint (S) would be half? Wait, maybe I made a mistake. Wait, the hypotenuse NM: NS is 6, SM is 2, so NM is 8? No, that can't be. Wait, the right triangle: leg RM is x, leg RN is y, hypotenuse NM is 6 + 2 = 8? Wait, no, the length from N to S is 6, S to M is 2, so NM is 8. Then, in a 30-60-90 triangle, the side opposite 30° is half the hypotenuse. Wait, angle at N is 30°, so the side opposite (RM) is half of NM? Wait, NM is 8, so RM would be 4? But that's not one of the options. Wait, no, maybe the hypotenuse is 8? Wait, no, the options for y: 4√3, 8, 2√3, 4. Wait, let's think again. The triangle is formed by folding an equilateral triangle, so the original equilateral triangle has side length equal to NM? Wait, no, when you fold an equilateral triangle along its altitude, you get a 30-60-90 triangle where the hypotenuse is the side of the equilateral triangle, the shorter leg is half of that, and the longer leg is (shorter leg)√3. Wait, maybe the length from N to M is 8? No, NS is 6, SM is 2, so NM is 8. Then, the longer leg (RN = y) would be (shorter leg)√3. Wait, shorter leg is 4 (half of 8), so longer leg is 4√3? Wait, 4√3 is one of the options. Wait, let's check: in 30-60-90 triangle, sides are \(a\), \(a\sqrt{3}\), \(2a\), where \(a\) is shorter leg, \(a\sqrt{3}\) is longer leg, \(2a\) is hypotenuse. If hypotenuse is 8 (2a = 8 ⇒ a = 4), then longer leg (y) is \(4\sqrt{3}\). Yes, that matches the first option.

Step2: Apply 30-60-90 Ratios

In a 30-60-90 triangle, hypotenuse \(= 2 \times \text{shorter leg}\), longer leg \(= \text{shorter leg} \times \sqrt{3}\). Here, hypotenuse \(NM = 6 + 2 = 8\) (since \(NS = 6\), \(SM = 2\), so total \(NM = 8\)). Thus, shorter leg (opposite 30°) is \(\frac{8}{2} = 4\). Longer leg (RN = y) is \(4 \times \sqrt{3} = 4\sqrt{3}\).

Answer:

\(4\sqrt{3}\) units (the first option: \(4\sqrt{3}\) units)