Question

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

Answer 1

Answer:

$d = 4\sqrt{3}$

$h=\dfrac{5\sqrt{2}}{2}$

Step-by-step explanation:

    When analyzing these triangles, we can notice something special about them. These triangles are the 30-60-90 and the 45-45-90 triangles. These triangles are special triangles that have specific ratios for their sides and angles.

    The right triangle on the left, for side d, is a 30-60-90 triangle. If the leg opposite of the 30-degree angle is x units long, then the leg opposite of the 60-degree angle is x√3 and the hypotenuse is 2x units long. With this information, we can solve for x. Then, we will find d.

         2x = 8

         x = 8 ÷ 2

         x = 4

         d = x√3

         d = 4√3

    The right triangle on the right, for side h, is a 45-45-90 triangle. The side lengths are always 1:1:√2, meaning that if one leg is x units long, then the other leg is also x units long, and the hypotenuse is x√2 units long. Keeping this in mind, we can solve for h. Then, we will rationalize the denominator.

         h√2 = 5

         $h=\dfrac{5}{\sqrt{2} }$

         $h=\dfrac{5*\sqrt{2}}{\sqrt{2} *\sqrt{2}}$

         $h=\dfrac{5\sqrt{2}}{2}$