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: 3/5 or 0.6
If x is the average (arithmetic mean) of m and 9, y is the average of 2m and 15, and z is the average of 3m and 18, what is the average of x, y, and z in terms of m?
A) m+6
B) m+7 ****
C) 2m+14
D) 3m+21
My answer ***
Are these correct ?
both are correct
Thanks !
ac length
To find the value of sinF in triangle DEF, we can use the information given about the similarity of the triangles and the length of the sides.
First, let's find the length of side EF in triangle DEF. Since each side of triangle DEF is 1/3 the length of the corresponding side in triangle ABC, we can calculate EF as:
EF = (1/3) * BC
= (1/3) * 16
= 16/3
Now, we know that triangle DEF is a right triangle since triangle ABC has a 90° angle at B. Therefore, we can use the Pythagorean theorem to find the length of side DF.
DF^2 = DE^2 + EF^2
Since DE corresponds to AC in triangle ABC and EF corresponds to BC, we can substitute the values:
DF^2 = AC^2 + BC^2
DF^2 = 20^2 + 16^2
DF^2 = 400 + 256
DF^2 = 656
Taking the square root of both sides:
DF = √656
Finally, to find the value of sinF, we can use the definition of sin which is opposite/hypotenuse. In triangle DEF, the side opposite angle F is EF and the hypotenuse is DF. Therefore:
sinF = EF/DF
sinF = (16/3)/√656
sinF = (16/3) * (1/√656)
sinF = (16/3) * (1/8√41)
sinF = 16/(3 * 8√41)
sinF = 2/(3√41)
sinF ≈ 0.169
So, the value of sinF is approximately 0.169 or you can simplify it further, sinF ≈ 1/6 or 0.167.
Therefore, the correct answer is not 3/5 or 0.6.