Question

question 2 (15 points)
based on the measures provided in the
diagram and that line \\(\overline{cb}\\) is tangent to the
circle, determine the measure of \\(\angle cdb\\).
(you may assume that point a is the center of the circle.)
(figure may not be drawn to scale.)
a) \\(38^\circ\\)
b) \\(71^\circ\\)
c) \\(76^\circ\\)
d) \\(109^\circ\\)

Explanation:

Step1: Find the central angle

The arc \( \overset{\frown}{BD} \) and the arc with measure \( 142^\circ \) are supplementary (since they form a semicircle? Wait, no, the total around a circle is \( 360^\circ \), but since \( A \) is the center, the straight line \( BE \) (wait, \( BD \) is a diameter? Wait, the tangent at \( D \), so \( AD \) is radius, and \( CD \) is tangent, so \( AD \perp CD \). Wait, the arc \( \overset{\frown}{B D} \) (the smaller arc) and the \( 142^\circ \) arc: the measure of the central angle for the smaller arc \( \overset{\frown}{BD} \) is \( 180^\circ - 142^\circ = 38^\circ \)? Wait, no, the tangent \( CD \) and the radius \( AD \) are perpendicular, so \( \angle CDA = 90^\circ \). Wait, maybe another approach. The inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. Wait, \( \angle CDB \) is an angle formed by a tangent and a chord. The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. The intercepted arc here is \( \overset{\frown}{BD} \). First, find the measure of arc \( \overset{\frown}{BD} \). The arc given is \( 142^\circ \), so the adjacent arc (the one not \( 142^\circ \)) is \( 360^\circ - 2\times142^\circ \)? No, wait, the diagram shows a circle with center \( A \), and \( BD \) is a chord, \( CD \) is tangent at \( D \). The arc from \( B \) to \( D \) (the minor arc) and the arc from \( B \) to \( D \) passing through the left side (142°). Wait, the total circumference is \( 360^\circ \), but if \( BD \) is a diameter, then the arc \( \overset{\frown}{BD} \) would be \( 180^\circ \), but here the arc on the left is \( 142^\circ \), so the arc \( \overset{\frown}{BD} \) (the minor arc) is \( 180^\circ - 142^\circ = 38^\circ \)? No, wait, the angle between tangent \( CD \) and chord \( BD \) is half the measure of the intercepted arc \( \overset{\frown}{BD} \). Wait, the formula is: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. So first, find the measure of arc \( \overset{\frown}{BD} \). The arc given is \( 142^\circ \), so the arc \( \overset{\frown}{BD} \) (the major arc? No, the minor arc) is \( 180^\circ - 142^\circ = 38^\circ \)? Wait, no, the tangent at \( D \), so the angle between tangent \( CD \) and chord \( BD \) is equal to half the measure of the intercepted arc \( \overset{\frown}{BD} \). Wait, the intercepted arc is the arc that is opposite the angle, i.e., the arc that is not containing the angle. Wait, the measure of \( \angle CDB \) is half the measure of the intercepted arc \( \overset{\frown}{BD} \). Wait, first, find the measure of arc \( \overset{\frown}{BD} \). The arc with measure \( 142^\circ \) and arc \( \overset{\frown}{BD} \) are supplementary? No, the total around the circle is \( 360^\circ \), but if \( BD \) is a diameter, then the arc \( \overset{\frown}{BD} \) is \( 180^\circ \), but the given arc is \( 142^\circ \), so the other arc (from \( B \) to \( D \) the short way) is \( 180^\circ - 142^\circ = 38^\circ \)? Wait, no, the tangent \( CD \) and the radius \( AD \) are perpendicular, so \( \angle CDA = 90^\circ \). Wait, maybe I made a mistake. Let's start over.

The tangent at \( D \), so \( AD \perp CD \) (tangent is perpendicular to radius at point of contact). So \( \angle CDA = 90^\circ \). Now, \( \angle CDB \) is the angle we need. Let's find \( \angle ADB \). \( \angle ADB \) is an inscribed angle? Wait, \( AD = AB \) (radii), so triangle \( ADB \) is isoceles? No, \( BD \) is a chord, \( AD \) and \( AB \) are radii. Wait, the arc \( \overset{\frown}{BD} \) has a central angle. The arc given is \( 142^\circ \), so the central angle for arc \( \overset{\frown}{BD} \) (the minor arc) is \( 180^\circ - 142^\circ = 38^\circ \)? Wait, no, the total around the circle is \( 360^\circ \), but if \( BD \) is a diameter, then the arc \( \overset{\frown}{BD} \) is \( 180^\circ \), but the arc on the left is \( 142^\circ \), so the arc \( \overset{\frown}{BD} \) (the minor arc) is \( 180^\circ - 142^\circ = 38^\circ \). Then, the inscribed angle over arc \( \overset{\frown}{BD} \) would be half, but \( \angle CDB \) is formed by tangent \( CD \) and chord \( BD \), so its measure is half the measure of the intercepted arc \( \overset{\frown}{BD} \). Wait, the formula is: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. So if the intercepted arc \( \overset{\frown}{BD} \) is \( 38^\circ \), then \( \angle CDB = \frac{1}{2} \times 38^\circ = 19^\circ \)? No, that's not one of the options. Wait, maybe the intercepted arc is the major arc. Wait, the arc given is \( 142^\circ \), so the major arc \( \overset{\frown}{BD} \) is \( 360^\circ - 142^\circ = 218^\circ \), but that can't be. Wait, maybe the arc is \( 142^\circ \) is the arc from \( B \) to \( D \) the long way, so the short way is \( 360^\circ - 2\times142^\circ \)? No, this is confusing. Wait, the options include \( 71^\circ \), \( 38^\circ \), etc. Let's look at the options. The correct answer is likely \( 71^\circ \). Wait, let's think again. The angle between tangent \( CD \) and chord \( BD \) is equal to half the measure of the intercepted arc \( \overset{\frown}{BD} \). The intercepted arc is the arc that is "cut off" by the chord \( BD \) and lies opposite the angle. So if the arc \( \overset{\frown}{B D} \) (the major arc) is \( 142^\circ \), then the minor arc is \( 360^\circ - 142^\circ = 218^\circ \)? No, that's not right. Wait, maybe the diagram has \( BD \) as a diameter, so the arc \( \overset{\frown}{B D} \) is \( 180^\circ \), and the given arc is \( 142^\circ \), so the other arc (from \( B \) to \( D \) the short way) is \( 180^\circ - 142^\circ = 38^\circ \). Then, the angle formed by tangent \( CD \) and chord \( BD \) is half the measure of the intercepted arc, which is the arc \( \overset{\frown}{B D} \) (the major arc? No, the formula is: the measure of the angle is half the measure of its intercepted arc. So if the angle is outside, wait, no: the angle between tangent and chord is equal to half the measure of the intercepted arc. So if the chord is \( BD \), the tangent is \( CD \), then the intercepted arc is \( \overset{\frown}{BD} \) (the arc that is between \( B \) and \( D \) and not containing \( C \)). Wait, maybe the arc given is \( 142^\circ \), so the arc \( \overset{\frown}{B D} \) (the one we need) is \( 180^\circ - 142^\circ = 38^\circ \), no, that's not. Wait, maybe the central angle for arc \( \overset{\frown}{B D} \) is \( 180^\circ - 142^\circ = 38^\circ \), then the inscribed angle over that arc would be \( 19^\circ \), but that's not an option. Wait, maybe I messed up the tangent angle formula. The correct formula is: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. So if the intercepted arc is \( 142^\circ \), then the angle is \( 71^\circ \). Ah! That makes sense. Wait, the arc \( \overset{\frown}{B D} \) (the one that's \( 142^\circ \)) is the intercepted arc? No, the angle formed by tangent \( CD \) and chord \( BD \) intercepts arc \( \overset{\frown}{B D} \) (the arc that's "inside" the angle). Wait, no, the 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 tangent is \( CD \) and the chord is \( BD \), then the intercepted arc is \( \overset{\frown}{B D} \) (the arc from \( B \) to \( D \) that's not containing \( C \)). Wait, maybe the arc given is \( 142^\circ \), so the intercepted arc is \( 142^\circ \), then the angle \( \angle CDB \) is \( \frac{1}{2} \times 142^\circ = 71^\circ \). Wait, but let's check the tangent and radius. \( AD \) is radius, \( CD \) is tangent, so \( \angle CDA = 90^\circ \). If \( \angle CDB = 71^\circ \), then \( \angle ADB = 90^\circ - 71^\circ = 19^\circ \). Then, the central angle for arc \( \overset{\frown}{B D} \) is \( 2 \times 19^\circ = 38^\circ \), which is supplementary to \( 142^\circ \) (38 + 142 = 180), which makes sense because they form a semicircle? Wait, no, 38 + 142 = 180, so they are supplementary, meaning they form a straight line (a diameter). So \( BD \) is a diameter, so the central angle for arc \( \overset{\frown}{B D} \) (the semicircle) is \( 180^\circ \), but here we have two arcs: 142° and 38°, which add up to 180°, so \( BD \) is a diameter. Then, the angle formed by tangent \( CD \) and chord \( BD \) is half the measure of the intercepted arc \( \overset{\frown}{B D} \) (the 142° arc). Wait, no, the intercepted arc is the one that's "cut off" by the chord and lies opposite the angle. So if the angle is at \( D \), between tangent \( CD \) and chord \( BD \), then the intercepted arc is \( \overset{\frown}{B D} \) (the arc from \( B \) to \( D \) that's not containing \( C \)), which is 142°, so the angle is \( \frac{1}{2} \times 142^\circ = 71^\circ \). That matches option b) 71°.

Step2: Apply 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. The intercepted arc here is \( \overset{\frown}{BD} \) with measure \( 142^\circ \). So \( \angle CDB = \frac{1}{2} \times 142^\circ = 71^\circ \).

Answer:

b) \( 71^\circ \)