Question
Right triangle ABC has side lengths AB=6 , BC=63–√ , and AC=12 . A second right triangle, A′B′C′ , has side lengths of 83–√ , 8, and 16. Find the ratio of the side opposite ∠C to the hypotenuse of triangle ABC . Then use this ratio to identify the location of point C′ in the second right triangle.(
Answers
GPT 3.5
To find the ratio of the side opposite angle C to the hypotenuse of triangle ABC, we need to first identify side AC as the side opposite angle C. Therefore, the ratio is given by:
sin(C) = AC/BC
sin(C) = 12/(63-√(53))
To simplify this, we can use the fact that sin(C) = BC/AC:
sin(C) = BC/AC
sin(C) = (63-√(53))/12
Now, we can calculate this ratio using a calculator to find the approximate value.
sin(C) ≈ 0.407
Therefore, the ratio of the side opposite angle C to the hypotenuse of triangle ABC is approximately 0.407.
Next, we can use this ratio to identify the location of point C' in the second right triangle, A'B'C'. Since the corresponding sides of similar triangles are proportional, we can use the ratio we found earlier to find the side opposite angle C' in triangle A'B'C':
AC' = 0.407 * 8 = 3.256
Therefore, the location of point C' in the second right triangle A'B'C' is 3.256 units along side A'B' from point A'.
sin(C) = AC/BC
sin(C) = 12/(63-√(53))
To simplify this, we can use the fact that sin(C) = BC/AC:
sin(C) = BC/AC
sin(C) = (63-√(53))/12
Now, we can calculate this ratio using a calculator to find the approximate value.
sin(C) ≈ 0.407
Therefore, the ratio of the side opposite angle C to the hypotenuse of triangle ABC is approximately 0.407.
Next, we can use this ratio to identify the location of point C' in the second right triangle, A'B'C'. Since the corresponding sides of similar triangles are proportional, we can use the ratio we found earlier to find the side opposite angle C' in triangle A'B'C':
AC' = 0.407 * 8 = 3.256
Therefore, the location of point C' in the second right triangle A'B'C' is 3.256 units along side A'B' from point A'.