Question

triangle mrn is created when an equilateral triangle is folded in half. what is the value of y? options: $2sqrt{3}$ units, 4 units, 8 units, $4sqrt{3}$ units

Explanation:

Step1: Determine the length of MN

Since triangle MRN is formed by folding an equilateral triangle in half, the length of MN should be equal to the length of MR (or RN, in the equilateral triangle). From the diagram, we see that MS = 2 and SN = 6, so MN = MS + SN = 2 + 6 = 8? Wait, no, actually, when folding an equilateral triangle in half, the side length of the equilateral triangle is MN. Wait, maybe I made a mistake. Wait, the right triangle RSN and RSM? Wait, no, the original triangle is equilateral, so when folded, MR = RN, and angle at R is 90? Wait, no, the equilateral triangle has all sides equal. Wait, maybe the length of MN is 8? Wait, no, let's re-examine. The segment MS is 2, SN is 6, so MN is 2 + 6 = 8? But in an equilateral triangle, all sides are equal. Wait, maybe the side length of the equilateral triangle is 8? Then, when folded, triangle MRN is a 30-60-90 triangle? Wait, no, the right angle is at R. Wait, maybe we can use the geometric mean or similar triangles. Wait, triangles RSM and RSN? Wait, no, triangle RSN is a right triangle? Wait, angle at S is right? Wait, the diagram shows right angles at R and S. Wait, maybe triangles RSM and NSR are similar? Wait, let's see. In right triangle MRN, with right angle at R, and RS is an altitude? Wait, no, maybe it's a 30-60-90 triangle. Wait, the length of MS is 2, SN is 6, so MN is 8? Wait, no, maybe the side length of the equilateral triangle is 8? Then, when folded, the height (y) can be found using Pythagoras. Wait, in a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. Wait, if MN is 8, then the side opposite 30 degrees would be 4, and the height (y) would be 4√3? Wait, no, maybe I messed up. Wait, let's start over.

Wait, the problem says triangle MRN is created when an equilateral triangle is folded in half. So the original equilateral triangle has a side length equal to MN. From the diagram, MS = 2, SN = 6, so MN = 2 + 6 = 8? Wait, no, that can't be. Wait, maybe MS is 2, and the original equilateral triangle has side length 8? Then, when folded, the height (y) of the right triangle MRN (which is a 30-60-90 triangle) can be calculated. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, where the hypotenuse is twice the shorter leg. Wait, if MN is 8 (hypotenuse), then the shorter leg (x) would be 4, and the longer leg (y) would be 4√3? Wait, no, wait. Wait, maybe the side length of the equilateral triangle is 8, so when folded, the base is 8, and the height (y) is the height of the equilateral triangle? Wait, the height of an equilateral triangle with side length s is (s√3)/2. If s = 8, then height is (8√3)/2 = 4√3? But that's not one of the options? Wait, no, the options are 2√3, 4, 8, 4√3. Wait, 4√3 is an option. Wait, maybe I made a mistake in MN length. Wait, maybe MS is 2, and SN is 6, so MN is 8? Then, in the right triangle MRN, with right angle at R, MR and RN are legs, and MN is hypotenuse? Wait, no, angle at R is right, so MR and RN are legs, MN is hypotenuse. Wait, but if it's folded from an equilateral triangle, then MR = RN? No, in an equilateral triangle, all sides are equal, so MR = MN = RN? Wait, that can't be. Wait, maybe the original equilateral triangle has side length 8, so MN = 8, and when folded, triangle MRN is a right triangle with legs y and x, and hypotenuse 8? No, that doesn't make sense. Wait, maybe the right angle is at S? Wait, the diagram shows right angles at R and S. Wait, maybe triangles RSM and NSR are similar. So RS² = MS * SN (geometric mean theorem). So RS² = 2 * 6 = 12, so RS = 2√3. Then, in triangle RSN, which is a right triangle, RN² = RS² + SN² = 12 + 36 = 48, so RN = 4√3. But y is RN? Wait, no, y is the length of RN? Wait, the diagram shows y as the length from R to N, horizontally. Wait, maybe y is RN, and since the original triangle is equilateral, MR = RN = 4√3? No, this is confusing. Wait, let's use the geometric mean. MS = 2, SN = 6, so RS = √(2*6) = √12 = 2√3. Then, in triangle RSN, RN² = RS² + SN² = (2√3)² + 6² = 12 + 36 = 48, so RN = √48 = 4√3. But the options include 4√3. Wait, but maybe y is RN? Wait, the diagram shows R to N as y, and R to M as x, and N to M as 8? Wait, no, MN is 8? Then, in right triangle MRN, with legs x and y, hypotenuse 8. But if RS is 2√3, and MS is 2, then x² = RS² + MS² = 12 + 4 = 16, so x = 4. Then, y² + x² = MN²? Wait, x is 4, MN is 8, so y² + 16 = 64, so y² = 48, y = 4√3. Ah, that makes sense. So y is 4√3.

Step2: Calculate y using Pythagoras

We found that x = 4 (from x² = 2² + (2√3)² = 4 + 12 = 16, so x = 4). Then, in right triangle MRN, with legs x = 4 and y, and hypotenuse MN = 8 (since MS = 2 and SN = 6, so MN = 2 + 6 = 8). Then, by Pythagoras:

y² + x² = MN²
y² + 4² = 8²
y² + 16 = 64
y² = 64 - 16
y² = 48
y = √48 = 4√3

Answer:

4√3 units (the option with 4√3 units)