Question

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.(1 point)
The length of the side opposite C′
is
units.
128 over 17 is not the right answer

Answer 1

To find the ratio of the side opposite \(\angle C\) to the hypotenuse of triangle \(ABC\), we first need to identify the sides:

- The side opposite \(\angle C\) is \(AB\), which has a length of \(8\).
- The hypotenuse is \(AC\), which has a length of \(17\).

Now we calculate the ratio:

\[
\text{Ratio} = \frac{AB}{AC} = \frac{8}{17}
\]

Next, in triangle \(A'B'C'\), we identify the sides:

- The hypotenuse (longest side) is likely \(A'C' = 34\).
- The side lengths are \(34\), \(30\), and \(16\).

The side opposite point \(C'\) (which corresponds to side length \(AB\) in triangle \(ABC\)) can be calculated using the same ratio \(\frac{8}{17}\).

Let \(x\) be the length of the side opposite \(C'\):

\[
\frac{x}{34} = \frac{8}{17}
\]

To find \(x\), we cross-multiply:

\[
17x = 8 \cdot 34
\]

Calculating \(8 \cdot 34\):

\[
8 \cdot 34 = 272
\]

Now, we solve for \(x\):

\[
17x = 272 \implies x = \frac{272}{17}
\]

Calculating \(272\div 17\):

\[
x = 16
\]

Therefore, the length of the side opposite \(C'\) is **16 units**.