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 Fcorrespond 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

Root3/2 is the answer

Answer 2

Answer: The value of sin F is √41/2.

Step-by-step explanation:

Since, triangles ABC and DEF are similar,

⇒ m∠ A = m∠ D, m∠B = m∠E and m∠C = m∠F

Also,

$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$

Here, m∠B = 90° ⇒ m∠E = 90°,

Now, BC=16, AC=20 ⇒ AB = √656 ( by Pythagoras theorem),

Since, each side of triangle DEF is 13 the length of the corresponding side of triangle ABC.

$\frac{\sqrt{656}}{DE}=\frac{20}{DF}=\frac{1}{13}$

⇒ DE = 13 × √656 = 52√41 , DF = 13 × 20 = 260,

Now, in triangle DEF,

$sin F = \frac{DE}{DF}=\frac{52\sqrt{41}}{260}=\frac{\sqrt{41}}{2}$