Question

a right triangle has a hypotenuse of length 16 and an angle of 45°, with a side opposite this angle of length $8sqrt{2}$. a second right triangle also has an angle of 45° and a side opposite this angle with a length of $4sqrt{2}$. determine the length of the hypotenuse in the second triangle. (1 point)

the hypotenuse of the second triangle has length 4.

the hypotenuse of the second triangle has length $8sqrt{2}$

the hypotenuse of the second triangle has length 8.

the hypotenuse of the second triangle has length $4sqrt{2}$

Explanation:

Step1: Recall sine - function formula

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For a $45^{\circ}$ angle, $\sin45^{\circ}=\frac{\sqrt{2}}{2}$.

Step2: Set up proportion using sine - function

In the first right - triangle, $\sin45^{\circ}=\frac{8\sqrt{2}}{16}=\frac{\sqrt{2}}{2}$. In the second right - triangle, let the hypotenuse be $h$. Since $\sin45^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}$, and the opposite side to the $45^{\circ}$ angle is $4\sqrt{2}$, we have $\sin45^{\circ}=\frac{4\sqrt{2}}{h}$.

Step3: Solve for the hypotenuse

Since $\sin45^{\circ}=\frac{\sqrt{2}}{2}$, we can set up the equation $\frac{\sqrt{2}}{2}=\frac{4\sqrt{2}}{h}$. Cross - multiply: $\sqrt{2}h = 2\times4\sqrt{2}$. Divide both sides by $\sqrt{2}$: $h = 8$.

Answer:

The hypotenuse of the second triangle has length 8.