right triangle ABC has side lengths AB=8, BC=15, and AC=17. A second right triangle, A'B'C, has side lengths of 34,30 and 16. First, find the ratio of the side opposite <C to the hypotenuse of triangle ABC. Then, use this ratio to identify the side length opposite point C' on the second right triangle.
To find the ratio of the side opposite angle C to the hypotenuse of triangle ABC, we need to find the length of the side opposite angle C and the length of the hypotenuse of triangle ABC.
Using the Pythagorean theorem, we can find the length of the side opposite angle C in triangle ABC:
AC^2 = AB^2 + BC^2
17^2 = 8^2 + 15^2
289 = 64 + 225
289 = 289
So the length of the side opposite angle C in triangle ABC is 17.
The length of the hypotenuse of triangle ABC is AC = 17.
The ratio of the side opposite angle C to the hypotenuse of triangle ABC is:
side opposite angle C / hypotenuse = 17/17 = 1/1 = 1.
To use this ratio to identify the side length opposite point C' on the second right triangle, we need to find the length of the hypotenuse of triangle A'B'C.
Using the Pythagorean theorem, we can find the length of the hypotenuse of triangle A'B'C:
A'C'^2 = A'B'^2 + B'C'^2
16^2 = 34^2 + 30^2
256 = 1156 + 900
256 = 2056
The length of the hypotenuse of triangle A'B'C is 16.
Now, using the ratio we found earlier, we can find the length of the side opposite point C' in triangle A'B'C:
side opposite point C' = ratio * hypotenuse
side opposite point C' = 1 * 16
side opposite point C' = 16.
So the side length opposite point C' on the second right triangle is 16.