Question

Find the measure of the arc or

angle indicated. Assume that
lines which appear tangent are
tangent.
O 35
O 75
O 60
O 65

Answer 1

60°

Explanation:

The measure of an angle formed by two tangents to a circle from a point outside the circle is one-half the difference between the intercepted arcs

You can see from the diagram that there are two arcs formed

We will call the arc farthest away from the angle as the far arc and that closest to the angle as a near arc

Since there are only two arcs, the far arc measure + near arc measure must be 360°

If we designate the measure of the far arc as Farc and measure of the near arc as Narc we get

Farc + Narc = 360°

We are given Farc = 240°

So
240° + Narc = 360°

Narc = 360 - 240 = 120°

Now we can compute m∠Q as follows

`m \angle Q = (1)/(2) \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle Q = (1)/(2) (240-120) \\\\ m \angle Q = (1)/(2) (120) \\\\ m \angle Q = 60^(\circ)`