Question

using algebra find the value of each variable. write answers in simplest radical form. 16. a right triangle with a 30° angle, hypotenuse 10, opposite side to 30° is c, adjacent side is d

Explanation:

Step1: Identify side roles in 30-60-90 triangle

In a 30-60-90 right triangle, the hypotenuse is twice the length of the shorter leg (opposite 30°). Here, hypotenuse = $10$, shorter leg = $c$.

Step2: Solve for $c$

Set up hypotenuse formula:
$10 = 2c$
$\frac{10}{2} = c$
$c = 5$

Step3: Solve for $d$ using Pythagoras

Use Pythagorean theorem: $a^2 + b^2 = c^2$, where hypotenuse $=10$, one leg $=c=5$, other leg $=d$.
$5^2 + d^2 = 10^2$
$25 + d^2 = 100$
$d^2 = 100 - 25 = 75$
$d = \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$

Answer:

$c=5$, $d=5\sqrt{3}$