Questions: For the right triangles below, find the exact values of the side lengths h and c. If necessary, write your responses in simplified radical form. h= c=

For the right triangles below, find the exact values of the side lengths h and c. If necessary, write your responses in simplified radical form.

h=

c=

Transcript text: For the right triangles below, find the exact values of the side lengths $h$ and $c$. If necessary, write your responses in simplified radical form. \[ h= \] \[ c= \]

Solution

Solution Steps

Step 1: Analyze the first triangle (45-45-90 triangle)

This is a 45-45-90 triangle, which means its sides are in the ratio $x:x:x\sqrt{2}$, where $x$ is the length of each leg and $x\sqrt{2}$ is the length of the hypotenuse. In this case, one leg is given as 8, so the other leg (h) is also 8.

Step 2: Calculate the hypotenuse of the first triangle.

The hypotenuse is $x\sqrt{2}$. Since $x=8$, the hypotenuse, $h$, is $8\sqrt{2}$.

Step 3: Analyze the second triangle (30-60-90 triangle)

This is a 30-60-90 triangle. The sides are in the ratio $x:x\sqrt{3}:2x$, where $x$ is the side opposite the 30° angle, $x\sqrt{3}$ is opposite the 60° angle, and $2x$ is the hypotenuse. We are given the side opposite the 30° angle as 4, so $x = 4$.

Step 4: Calculate the length of side c

Side c is opposite the 60° angle, so its length is $x\sqrt{3}$. Since $x=4$, the length of side c is $4\sqrt{3}$.