Question

1. point m lies in the interior of ∠geo. if (mangle gem=\frac{2}{3}x^{circ},mangle meo = \frac{1}{4}x^{circ}), and (mangle geo=\frac{11}{x}^{circ}), find the value of x.

Explanation:

Step1: Use angle - addition postulate

Since point $M$ lies in the interior of $\angle GEO$, we have $m\angle GEM + m\angle MEO=m\angle GEO$. So, $\frac{2}{3}x+\frac{1}{4}x = \frac{11}{x}$.

Step2: Find a common denominator for the left - hand side

The common denominator of 3 and 4 is 12. Rewrite the left - hand side as $\frac{2\times4}{3\times4}x+\frac{1\times3}{4\times3}x=\frac{8}{12}x+\frac{3}{12}x=\frac{8x + 3x}{12}=\frac{11x}{12}$.

Step3: Solve the resulting equation

We now have the equation $\frac{11x}{12}=\frac{11}{x}$. Cross - multiply to get $11x\times x=11\times12$, which simplifies to $11x^{2}=132$. Divide both sides by 11: $x^{2}=12$. Then $x=\pm\sqrt{12}=\pm2\sqrt{3}$. But since $x$ represents an angle measure (in a non - negative context in terms of the angle formula), we take $x = 2\sqrt{3}$.

Answer:

$x = 2\sqrt{3}$