Question

special right triangles quick check 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? (1 point) 36 inches 18√3 inches 18 inches 9 inches

Explanation:

Step1: Analyze the equilateral triangle

An equilateral triangle can be split into two 30 - 60 - 90 right triangles by its perpendicular bisector (which is also the altitude, median, and angle bisector). The side of the equilateral triangle (36 inches) becomes the hypotenuse of each 30 - 60 - 90 triangle, and the base of each right triangle is half of 36 inches, so \(\frac{36}{2}=18\) inches.

Step2: Use 30 - 60 - 90 triangle ratios

In a 30 - 60 - 90 right triangle, the ratios of the sides are \(1:\sqrt{3}:2\) (opposite 30°, 60°, and 90° respectively). Here, the side opposite 30° is 18 inches (the shorter leg), the side opposite 60° is the perpendicular bisector (altitude) we want to find, and the hypotenuse is 36 inches. If the shorter leg (opposite 30°) is \(x\), the longer leg (opposite 60°) is \(x\sqrt{3}\). Since \(x = 18\), the longer leg (perpendicular bisector) is \(18\sqrt{3}\) inches.

Answer:

\(18\sqrt{3}\) inches (corresponding to the option: \(18\sqrt{3}\) inches)