In the figure above, . If and , what is the length of ?

Question

Accepted Answer

The correct answer is 6. Since , and are both similar to 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since , which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of to the 3-4-5 triangle is 3:1. Therefore, the length of (the side opposite to ) is , and the length of (the side adjacent to ) is . It is also given that . Since and , it follows that , which means that the similarity ratio of to the 3-4-5 triangle is 2:1 ( is the side adjacent to ). Therefore, the length of , which is the side opposite to , is .

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