Use what you know about the relationships in 30-60-90 right triangles to solve the following problem. A stained-glass window is in the shape of an equilateral triangle with sides that are 36 inches long. How long is the perpendicular bisector of any side?
In a 30-60-90 right triangle, the sides are always in the ratio 1:√3:2.
In an equilateral triangle, each interior angle measures 60 degrees.
If we draw the perpendicular bisector of any side of the equilateral triangle, we will have two congruent right triangles, each with one leg measuring half the length of the side of the equilateral triangle.
Let the length of the perpendicular bisector be x.
Since the length of the side of the equilateral triangle is 36 inches, the length of one of the legs of the right triangle is 36/2 = 18 inches.
Using the ratios of 30-60-90 right triangles, we know that the ratio of the length of the shortest leg to the length of the hypotenuse is 1:2.
Therefore, we can set up the following equation to solve for x:
18/x = 1/2
Cross multiplying:
2 * 18 = x * 1
36 = x
Therefore, the length of the perpendicular bisector is 36 inches.