Question

In triangle ABC, the measure of ∠B is 90°, BC=16, and AC=20. Triangle DEF is similar to triangle ABC, where vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side of triangle DEF is 13 the length of the corresponding side of triangle ABC. What is the value of sinF?

Answer 1

Sin F = 3/5

Explanation:

Since F= C because the triangles are similar

Sin F = Sin C since the triangles are similar

sin C = opposite / hypotenuse

=AB/AC

We know AC =20, but we need to determine AB

Since this is a right triangle

AB^2 + BC^2 = AC^2

AB^2 +16^2 = 20^2

AB^2 + 256=400

Subtract 256 from each side

AB^2 +256-256 = 400-256

AB^2 =144

Take the square root of each side

sqrt(AB^2) =sqrt(144)

AB = 12

sin C = 12/20

Divide the top and bottom by 4

sin C = 3/5

Sin F = 3/5

Answer 2

You are given BC and AC, lets find AB by using the Pythagorean theorem:

AB = √(20^2 - 16^2)

AB = √(400-256)

AB = √144

AB = 12

Since the triangles are similar, sin F would be the same as sin C

SinC = Opposite side / hypotenuse

SinC = 12/20, reduces to 3/5

SinF = SinC = 3/5

Then if you need the angle of F:

F = arcsin(3/5) = 36.87 degrees.