Question in-the
In the figure above, 
. If 
and 
, what is the length of 
?
In the figure above, . If and , what is the length of ?
Hard-difficulty · SAT Math · Right Triangles & Trigonometry — Trig ratios (sin, cos, tan) in right triangles. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.
Answer explanation
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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