When an equilateral triangle is folded in half, it creates two congruent right triangles. In the case of triangle MRN, the folding creates a right triangle where the original triangle’s base becomes one leg of the right triangle, and the height forms the other leg. The hypotenuse is the original side of the equilateral triangle.
To find the value of y, we need to remember that in an equilateral triangle, all sides are equal and each angle is 60 degrees. When the triangle is folded in half, it forms a right angle (90 degrees) between the height and half of the base.
Let’s denote the side length of the equilateral triangle as ‘s’. The height (h) of the triangle can be calculated using the formula:
h = (√3/2)s
After folding, the base of the triangle is now s/2, and the height is h as calculated above.
Using the Pythagorean theorem in triangle MRN:
y² = (s/2)² + h²
By substituting the height formula:
y² = (s/2)² + (√3/2)² s²
Simplifying this:
y² = (s²/4) + (3s²/4) = s²
Thus, we find that:
y = s
In conclusion, the value of y corresponds to the length of one side of the equilateral triangle. So, if the side length of the triangle is known, that same length will also be the value of y after the triangle has been folded in half.