Question

what is the measure of arc qsr?
diagram of a circle with center, points q, s, r on the circle, and a tangent from an external point forming a 70° angle at the external point with the two tangents to q and r

Explanation:

Step1: Recall tangent properties

A tangent to a circle is perpendicular to the radius at the point of contact. So, $\angle OQR = 90^\circ$ and $\angle ORQ = 90^\circ$, where $O$ is the center.

Step2: Find central angle $\angle QOR$

In quadrilateral $OQPR$ (let $P$ be the external point), the sum of interior angles is $360^\circ$. We know two right angles ($90^\circ$ each) and the external angle $70^\circ$. So, $\angle QOR = 360^\circ - 90^\circ - 90^\circ - 70^\circ = 110^\circ$.

Step3: Calculate arc QSR

The total circumference of a circle corresponds to $360^\circ$. Arc QSR is the major arc, so its measure is $360^\circ - \angle QOR$. Substituting $\angle QOR = 110^\circ$, we get $360^\circ - 110^\circ = 250^\circ$.

Answer:

$250$