Question

Plz i need help

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 1/3 the length of the corresponding side of triangle ABC. What is the value of sinF?

Answer 1

Sin F = 0.6

Explanation:

From triangle ABC, applying Pythagoras theorem to determine the length AB;

`/hyp/^(2)` =
`/adj1/^(2)` +
`/adj2/^(2)`

`/AC/^(2)` =
`/BC/^(2)` +
`/AB/^(2)`

`/20/^(2)` =
`/16/^(2)` +
`/AB/^(2)`

400 = 256 +
`/AB/^(2)`

`/AB/^(2)` = 400 - 256

= 144

AB =
`√(144)`

= 12

AB = 12, AC = 20, BC = 16

Therefore since ΔABC ≅ ΔDEF, and each side of triangle DEF is
`(1)/(3)` the length of the corresponding side of triangle ABC.

Then,

DE =
`(1)/(3)`AB =
`(1)/(3)` x 12

= 4

EF =
`(1)/(3)`BC =
`(1)/(3)` x 16

=
`(16)/(3)`

DF =
`(1)/(3)`AC =
`(1)/(3)` x 20

=
`(20)/(3)`

Then, applying trigonometric function to ΔDEF, we have;

Sin F =
`(opposite)/(hypotenuse)`

=
`(DE)/(DF)`

= 4 ÷
`(20)/(3)`

= 4 x
`(3)/(20)`

=
`(3)/(5)`

Sin F = 0.6