Question

what is the measure of ∠qap? diagram: circle with points q, p, a, o; arc labeled 110°; multiple-choice options: a. 40°, b. 60°, c. 25°, d. 55°

Explanation:

To determine the measure of \(\angle QAE\), we use the property of the angle formed by a tangent and a chord (or secant) with the intercepted arc. The measure of an angle formed outside the circle is half the difference of the measures of the intercepted arcs. However, in this case, we can also use the fact that the measure of an inscribed angle is half the measure of its intercepted arc, and for angles formed by a tangent and a chord, the angle is half the measure of the intercepted arc.

Looking at the diagram, the intercepted arc \(QE\) is \(110^\circ\). Wait, actually, the measure of the angle formed by a tangent and a chord is half the measure of the intercepted arc. Wait, maybe the arc opposite to the angle? Wait, let's re - examine.

Wait, the total circumference of a circle is \(360^\circ\), but the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, if the arc \(QE\) is \(110^\circ\), then the adjacent arc (the major arc or minor arc) - Wait, maybe the angle \(\angle QAE\) is an angle formed by a tangent \(AQ\) and a chord \(AE\). The measure of \(\angle QAE\) should be half the measure of the intercepted arc \(QE\). Wait, no, if the arc \(QE\) is \(110^\circ\), then the angle formed by the tangent and the chord is half the measure of the intercepted arc. Wait, but maybe the other arc: the total around the circle is \(360^\circ\), but the angle between tangent and chord is half the measure of the intercepted arc. Wait, let's assume that the arc \(QE\) is \(110^\circ\), then the angle \(\angle QAE=\frac{1}{2}\times(180 - 110)\)? No, that's not right. Wait, the correct formula is: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the intercepted arc (the arc that is "cut off" by the chord and the tangent) is \(110^\circ\), then \(\angle QAE=\frac{1}{2}\times110 = 55^\circ\)? Wait, no, maybe the arc is \(110^\circ\), and the angle is half of the difference? Wait, no, the formula for the angle formed by a tangent and a chord is: \(m\angle=\frac{1}{2}m(\text{intercepted arc})\).

Looking at the options, if the intercepted arc is \(110^\circ\), then \(\frac{1}{2}\times110 = 55^\circ\), which is option D.

Step1: Recall the tangent - chord angle theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Step2: Identify the intercepted arc

From the diagram, the intercepted arc (the arc between the chord and the tangent) has a measure of \(110^\circ\).

Step3: Calculate the angle

Using the formula \(m\angle QAE=\frac{1}{2}\times m(\text{intercepted arc})\), we substitute \(m(\text{intercepted arc}) = 110^\circ\). So \(m\angle QAE=\frac{1}{2}\times110^\circ = 55^\circ\).

Answer:

D. \(55^\circ\)