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 45 - 45 - 90 triangle part:

In the 45 - 45 - 90 triangle with hypotenuse , using the ratio hypotenuse , where  is the length of a leg. Let the leg length be . Then , solving for  gives

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