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Right triangles and trigonometry SAT question
Math · Geometry and Trigonometry · Hard · Answer it here, explanation included.
In the figure above, 
. If 
and 
, what is the length of 
?
Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.
ExplanationShow
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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