Question

1. using pythagoreans theorem and exact answers, determine the value of x in each triangle. 11) 12) 13) 14) 16) what do the triangles in problems 10 - 15 all have in common? 17) use the pattern you saw in problems 10 - 15 to generate the blank spaces of this triangle with side length x. 18) the triangle. blanks.

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the side opposite the 30 - degree angle is $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

Step2: Identify given side

Given the side opposite the 60 - degree angle is $x$. Let the side opposite the 30 - degree angle be $y$ and the hypotenuse be $z$.

Step3: Find side opposite 30 - degree angle

Since the side opposite 60 - degree angle $x = y\sqrt{3}$, then $y=\frac{x}{\sqrt{3}}=\frac{\sqrt{3}x}{3}$.

Step4: Find the hypotenuse

The hypotenuse $z = 2y$. Substituting $y=\frac{\sqrt{3}x}{3}$, we get $z=\frac{2\sqrt{3}x}{3}$.

Answer:

The side opposite the 30 - degree angle is $\frac{\sqrt{3}x}{3}$ and the hypotenuse is $\frac{2\sqrt{3}x}{3}$