Question

In $△ABC$, the measure of $\angle B$ is ${90}^{\circ },BC=16$, and $AC=20$. $△DEF$ is similar to $△ABC$, where vertices $D,E,$ and $F$ correspond to vertices. $A,B$, and $C$, respectively, and each side of $△DEF$ is $\frac{1}{3}$ the length of the corresponding side of $△ABC$. What is the value of $\sin F$?

• A. $\frac{3}{5}$
• B. $\frac{5}{3}$
  
• C. $-\frac{3}{5}$
• D. $-\frac{5}{3}$

Answer 1

The correct option is A $\frac{3}{5}$

$△ABC$ is a right triangle with its right angle at $B$.

Thus, $\overline{AC}$ is the hypotenuse of right triangle $ABC$, and$\overline{AB}$ and$\overline{BC}$ are perpendicular to each other.

By the Pythagorean theorem,

$AB=\sqrt{{20}^2-{16}^2}=\sqrt{400-256}=\sqrt{144}=12$

Given: $△DEF\sim △ABC$

And, $\angle B=\angle E={90}^o$

Thus, $\sin F=\sin C$

From the side lengths of $△ABC$,

$\sin C=\frac{AB}{AC}=\frac{12}{20}=\frac{3}{5}$

Therefore, $\sin F=\frac{3}{5}$