Question

in circle $m$, $m\angle lon = 57^{\circ}$. solve for $x$ if $m\overset{\frown}{ln} = (10x - 25)^{\circ}$. if necessary, round your answer to the nearest tenth.

Explanation:

Step1: Relate angle and arc measure

The measure of an inscribed angle is half the measure of its intercepted arc. So:
$$m\angle LON = \frac{1}{2}m\overset{\frown}{LN}$$

Step2: Substitute given values

Plug in $m\angle LON=57^\circ$ and $m\overset{\frown}{LN}=(10x-25)^\circ$:
$$57 = \frac{1}{2}(10x - 25)$$

Step3: Eliminate the fraction

Multiply both sides by 2:
$$57 \times 2 = 10x - 25$$
$$114 = 10x - 25$$

Step4: Isolate the variable term

Add 25 to both sides:
$$114 + 25 = 10x$$
$$139 = 10x$$

Step5: Solve for x

Divide both sides by 10:
$$x = \frac{139}{10} = 13.9$$

Answer:

$13.9$