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?
Answers
Step-by-step explanation:
Triangle ABC is a right triangle with its right angle at B. Therefore, AC is the hypotenuse of right triangle ABC, and AB and BC are the legs of right triangle ABC. According to the Pythagorean theorem,
AB=√202−162=√400−256=√144=12
Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of angle∠F equals the measure of angle∠C. Therefore, sinF=sinC. From the side lengths of triangle ABC,
sinF=
oppositeside
hypotenuse
=
AB
AC
=
12
20
=
3
5
Therefore, sinF=
3
5
.
The final answer is
3
5
or 0.6.
Answer:
△ABC is a right triangle with its right angle at B.
Thus,
AC
is the hypotenuse of right triangle ABC, and
AB
and
BC
are perpendicular to each other.
By the Pythagorean theorem,
AB=
20
2
−16
2
=
400−256
=
144
=12
Given: △DEF∼△ABC
And, ∠B=∠E=90
o
Thus, sinF=sinC
From the side lengths of △ABC,
sinC=
AC
AB
=
20
12
=
5
3
Therefore, sinF=
5
3