Question

finding arc measures involving two intersecting tangents
what is the measure of arc qsr?
70°

Explanation:

Step1: Recall tangent angle theorem

The measure of an angle formed by two tangents outside a circle is half the difference of the measures of the intercepted arcs. Let the measure of arc $QR$ be $x$, and arc $QSR$ be $y$. The theorem gives:
$$70^\circ = \frac{1}{2}(y - x)$$

Step2: Total circle arc measure

The sum of arcs in a circle is $360^\circ$, so:
$$x + y = 360^\circ \implies x = 360^\circ - y$$

Step3: Substitute $x$ into the equation

Replace $x$ in the first equation:
$$70^\circ = \frac{1}{2}(y - (360^\circ - y))$$

Step4: Simplify and solve for $y$

Expand and isolate $y$:
$$70^\circ = \frac{1}{2}(2y - 360^\circ)$$
$$70^\circ = y - 180^\circ$$
$$y = 70^\circ + 180^\circ = 250^\circ$$

Answer:

$250$