Question

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

Explanation:

Step1: Recall the formula for angle formed by two tangents

The measure of the angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs. Let the measure of arc QSR be $x$ and the measure of the other arc be $y$. The angle between the tangents is $70^{\circ}$, and we know that $x + y=360^{\circ}$ (a full - circle). Also, the formula for the angle between the tangents $\theta=\frac{1}{2}(x - y)$.

Step2: Substitute the known values

Since $\theta = 70^{\circ}$, we have $70=\frac{1}{2}(x - y)$. Multiply both sides by 2 to get $140=x - y$. And we know $y = 360 - x$. Substitute $y$ into the equation $140=x-(360 - x)$.

Step3: Solve the equation

Expand the right - hand side: $140=x - 360+x$. Combine like terms: $140 = 2x-360$. Add 360 to both sides: $140 + 360=2x$, so $500 = 2x$. Divide both sides by 2: $x = 250$.

Answer:

$250$