Geometry

Question Analysis

This question focuses on using the properties of 45 - 45 - 90 and 30 - 60 - 90 right - triangles to find the value of . The key is applying the side - length relationships specific to these special triangles.

 

Key Concept Explanation

45 - 45 - 90 triangle: In a 45 - 45 - 90 triangle, if the leg has length , the hypotenuse has length .

30 - 60 - 90 triangle: In a 30 - 60 - 90 triangle, if the short leg (opposite 30°) has length , the long leg (opposite 60°) has length , and the hypotenuse (opposite 90°) has length .

 

Step - by - Step Solution

1. Analyze the 30 - 60 - 90 triangle part:

The side with length  is the hypotenuse of the 30 - 60 - 90 triangle. Let the short leg of this 30 - 60 - 90 triangle be . Using the formula hypotenuse , and given the hypotenuse is , we solve for :

The long leg of the 30 - 60 - 90 triangle (opposite the 60° angle) is . Substitute