Question

Right triangle ABC has side lengths AB=21 , BC=28 , and AC=35 . Find the ratio of the side opposite ∠C to the hypotenuse of triangle ABC . Express your ratio as a decimal.(1 point) The ratio of the side opposite ∠C to the hypotenuse is

Answer 1

In triangle \(ABC\), we identify the sides as follows:
- \(AB = 21\)
- \(BC = 28\)
- \(AC = 35\)

To confirm that triangle \(ABC\) is indeed a right triangle, we will check if the Pythagorean theorem holds:

\[
AB^2 + BC^2 = AC^2
\]

Calculating the squares of the side lengths:

\[
AB^2 = 21^2 = 441
\]
\[
BC^2 = 28^2 = 784
\]
\[
AC^2 = 35^2 = 1225
\]

Now, we check the Pythagorean theorem:

\[
AB^2 + BC^2 = 441 + 784 = 1225
\]
\[
AC^2 = 1225
\]

Since \(AB^2 + BC^2 = AC^2\), triangle \(ABC\) is a right triangle with the hypotenuse being \(AC\).

Next, we are asked to find the ratio of the side opposite \(\angle C\) to the hypotenuse of triangle \(ABC\). The side opposite \(\angle C\) is \(AB\) (21) and the hypotenuse is \(AC\) (35).

Thus, the ratio can be expressed as:

\[
\text{Ratio} = \frac{AB}{AC} = \frac{21}{35}
\]

To simplify this fraction:

\[
\frac{21}{35} = \frac{3}{5}
\]

Now, we convert the simplified ratio to a decimal:

\[
\frac{3}{5} = 0.6
\]

Therefore, the ratio of the side opposite \(\angle C\) to the hypotenuse is

\[
\boxed{0.6}
\]