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?

11 answers

Since ∠B is a right angle, we have sin(B) = 1 and cos(B) = 0.

Using the Pythagorean theorem, we can find the length of the third side of triangle ABC:

AB^2 = AC^2 - BC^2 = 20^2 - 16^2 = 144

So AB = 12.

Now, since triangles DEF and ABC are similar, we know that the corresponding angles are equal. In particular, we have

∠F = ∠C

and

∠E = ∠B

Since ∠B is a right angle, we have ∠E = 90°.

Using the fact that the sides of triangle DEF are 1/3 the length of the corresponding sides of triangle ABC, we have

DE = 1/3 AB = 4

and

EF = 1/3 BC = 16/3

Now we can use the Pythagorean theorem to find DF:

DF^2 = DE^2 + EF^2 = 4^2 + (16/3)^2 = 256/9

So DF = √(256/9) = 16/3.

Finally, we can use the sine function to find sin(F):

sin(F) = DF/EF = (16/3) / (16/3) = 1.

Therefore, sin(F) = 1.

Thank you.

You're welcome!

The final answer is
3/5
or 0.6.

I apologize for my mistake earlier. You are correct, the final answer should be sin(F) = 3/5 or 0.6. Thank you for bringing it to my attention.

welcome.

Let me know if you have any further questions or if there is anything else I can help you with!

alright

Great! Have a good day!

you too

Thank you!