Question

in circle $n$, $m\angle mpo = 63^\circ$. solve for $x$ if $m\overset{\frown}{mo} = (5x - 32)^\circ$. if necessary, round your answer to the nearest tenth.

Explanation:

Step1: Apply inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Here, $\angle MPO$ is an inscribed angle intercepting arc $\widehat{MO}$, so:
$$\text{m}\angle MPO = \frac{1}{2} \text{m}\widehat{MO}$$

Step2: Substitute given values

Plug in $\text{m}\angle MPO = 63^\circ$ and $\text{m}\widehat{MO} = (5x - 32)^\circ$:
$$63 = \frac{1}{2}(5x - 32)$$

Step3: Solve for $x$ (isolate term with $x$)

Multiply both sides by 2:
$$126 = 5x - 32$$

Step4: Isolate $5x$

Add 32 to both sides:
$$126 + 32 = 5x$$
$$158 = 5x$$

Step5: Solve for $x$

Divide both sides by 5:
$$x = \frac{158}{5} = 31.6$$

Answer:

$x = 31.6$