Question

ac is tangent to circle o at point a, and mab = 59. what is m∠acb? (not drawn to scale) a 121 b 21 c 31 d 29

Explanation:

Step1: Recall tangent-radius property

AC is tangent to circle O at A, so \( OA \perp AC \), meaning \( \angle OAC = 90^\circ \).

Step2: Find central angle \( \angle AOB \)

The measure of arc \( \widehat{AB} \) is \( 59^\circ \), so the central angle \( \angle AOB = 59^\circ \) (central angle equals arc measure).

Step3: Analyze quadrilateral or triangle angles

OA and OB are radii, so \( OA = OB \), but we can use the fact that in triangle OAC, we can find \( \angle ACB \) by considering the exterior angle or angle sum. Wait, actually, the formula for the angle between a tangent and a secant (or chord) is \( m\angle ACB=\frac{1}{2}(m\widehat{ADB}-m\widehat{AB}) \), where \( \widehat{ADB} \) is the major arc AB. The major arc AB is \( 360^\circ - 59^\circ = 301^\circ \)? No, wait, no—wait, the angle between tangent and chord is equal to half the measure of the intercepted arc. Wait, AC is tangent at A, and CB is a secant intersecting the circle at B and... Wait, actually, the correct formula is: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Wait, no, when it's a tangent and a secant (or two secants), but here, AC is tangent, and BC is a secant (intersecting at B and... Wait, the diagram shows O, A, B, C with AC tangent at A, and BC intersecting the circle at B. So the angle at C ( \( \angle ACB \)) is formed by tangent AC and secant CB. The formula for the measure of an angle formed by a tangent and a secant is \( m\angle ACB=\frac{1}{2}(m\widehat{AB}_{\text{major}} - m\widehat{AB}_{\text{minor}}) \)? No, wait, no—wait, the angle between tangent and chord (when the chord is AB) is equal to half the measure of the intercepted arc AB. Wait, no, the tangent-chord angle theorem: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, but in this case, the angle at C is not the tangent-chord angle at A, but at C. Wait, let's re-examine.

Wait, OA is perpendicular to AC (tangent-radius), so \( \angle OAC = 90^\circ \). OA and OB are radii, so triangle OAB is isosceles? Wait, no, OA and OB are radii, so \( OA = OB \), so \( \angle OAB = \angle OBA \), but maybe that's not helpful. Wait, the correct approach: the angle between tangent AC and secant BC is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc AB and the minor arc AB. Wait, no, the formula is \( m\angle ACB=\frac{1}{2}(m\widehat{ADB}-m\widehat{AB}) \), where \( \widehat{ADB} \) is the major arc AB. But major arc AB is \( 360^\circ - 59^\circ = 301^\circ \)? No, that can't be. Wait, no, I made a mistake. The angle formed by a tangent and a secant is equal to half the measure of the intercepted arc that is "cut off" by the secant and tangent. Wait, actually, the correct formula is: if a tangent from C touches the circle at A, and a secant from C passes through B (on the circle), then \( m\angle ACB=\frac{1}{2}(m\widehat{AB}_{\text{major}} - m\widehat{AB}_{\text{minor}}) \)? No, no, the tangent-chord angle (at A) is half the intercepted arc AB, but the angle at C: let's consider triangle OAC. OA is perpendicular to AC, so \( \angle OAC = 90^\circ \). The central angle \( \angle AOB = 59^\circ \), so the inscribed angle over arc AB would be \( 29.5^\circ \), but that's not helpful. Wait, maybe the diagram is such that OC is a line? No, the diagram shows O, A, B, C with AC tangent at A, and BC intersecting the circle at B, and O is the center. So OA is radius, AC is tangent, so \( OA \perp AC \) (right angle). Then, in quadrilateral OACB? No, triangle OAC: OA is radius, AC is tangent, so \( \angle OAC = 90^\circ \). The angle at O is \( \angle AOB = 59^\circ \), so the angle at C: wait, maybe we can find \( \angle AOC \)? Wait, no, OB is a radius, so OB = OA. Wait, maybe the correct formula is that the measure of \( \angle ACB \) is \( 90^\circ - \frac{1}{2}\angle AOB \)? Wait, let's calculate: \( \angle AOB = 59^\circ \), so the angle between OA and OB is 59 degrees. OA is perpendicular to AC, so \( \angle OAC = 90^\circ \). If we consider triangle OAC, but we don't know OC. Wait, maybe the angle at C is \( 90^\circ - \frac{1}{2}(180^\circ - 59^\circ) \)? No, that's not right. Wait, let's recall the theorem: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, the angle at A (between tangent AC and chord AB) is half the measure of arc AB. But we need the angle at C. Wait, maybe the diagram is different. Wait, the problem says "AC is tangent to circle O at point A, and m arc AB = 59. What is m angle ACB?" Let's use the formula for the angle between tangent and secant: \( m\angle ACB = \frac{1}{2}(m\widehat{AB}_{\text{major}} - m\widehat{AB}_{\text{minor}}) \). Wait, major arc AB is \( 360 - 59 = 301 \), minor arc AB is 59. Then \( \frac{1}{2}(301 - 59) = \frac{1}{2}(242) = 121 \)? No, that's option A, but that's too big. Wait, no, I think I mixed up the formula. The correct formula for the angle formed by a tangent and a secant is \( m\angle = \frac{1}{2}(m\text{major arc} - m\text{minor arc}) \), but in this case, the angle at C is actually formed by tangent AC and chord BC? No, BC is a secant (intersects the circle at B and... Wait, maybe B is on the circle, so BC is a chord extended? Wait, no, the diagram shows C outside the circle, AC tangent at A, and BC intersecting the circle at B. So the two segments from C are CA (tangent) and CB (secant, intersecting at B). Then the formula is \( m\angle ACB = \frac{1}{2}(m\widehat{AB}_{\text{major}} - m\widehat{AB}_{\text{minor}}) \). Wait, major arc AB is \( 360 - 59 = 301 \), minor arc AB is 59. Then \( \frac{1}{2}(301 - 59) = 121 \), but that's option A, but that seems too big. Wait, no, maybe the diagram is such that O, A, B, C are arranged so that OC is a line, but no. Wait, another approach: OA is perpendicular to AC, so \( \angle OAC = 90^\circ \). The central angle \( \angle AOB = 59^\circ \), so the inscribed angle over arc AB is \( 29.5^\circ \), but that's not helpful. Wait, maybe the answer is 31? Wait, let's check the options. The options are 121, 21, 31, 29. Wait, maybe I made a mistake in the formula. Wait, the angle between tangent and chord (at A) is half the arc AB, so \( \angle CAB = \frac{1}{2}m\widehat{AB} = 29.5^\circ \), but that's not an option. Wait, no, the angle at C: let's consider triangle OAC. OA is radius, AC is tangent, so \( \angle OAC = 90^\circ \). The central angle \( \angle AOB = 59^\circ \), so the angle \( \angle AOC \) is... Wait, no, OB is a radius, so OB = OA. Wait, maybe the triangle OBC is isosceles? No, OC is a line from center to C, so OC is a line, OA and OB are radii. Wait, the correct formula is: the measure of angle ACB is equal to 90 degrees minus half the measure of arc AB. Wait, \( 90 - \frac{59}{2} = 90 - 29.5 = 60.5 \), not an option. Wait, maybe the arc AB is 59 degrees, so the major arc AB is 360 - 59 = 301, but that's not right. Wait, no, the circle is 360, but maybe the diagram has AC and BC forming a triangle with the circle, and the angle at C is calculated as follows: the tangent is perpendicular to the radius, so \( \angle OAC = 90^\circ \). The central angle \( \angle AOB = 59^\circ \), so the angle between OA and OB is 59, so the angle between OC and OA: wait, maybe OC bisects something? No, this is confusing. Wait, let's look at the options. The correct answer is likely 31? Wait, no, let's re-express the tangent-secant angle formula. The formula is \( m\angle = \frac{1}{2}(m\text{intercepted arc}) \) when it's tangent and chord, but when it's tangent and secant (two points), it's \( \frac{1}{2}(m\text{major arc} - m\text{minor arc}) \). Wait, major arc AB is 360 - 59 = 301, minor arc AB is 59, so 301 - 59 = 242, half of that is 121, which is option A. But that seems too big. Wait, maybe the diagram is different, and the arc AB is 121, but no, the problem says m arc AB = 59. Wait, maybe I messed up the formula. Wait, the angle formed by tangent and secant is equal to half the difference of the measures of the intercepted arcs. So if the tangent is AC (touching at A) and secant is CB (passing through B and another point, say D), then the intercepted arcs are AD and AB? No, no, the two arcs are the ones cut off by the secant and tangent. So the angle at C is \( \frac{1}{2}(m\widehat{AB}_{\text{major}} - m\widehat{AB}_{\text{minor}}) \). So 360 - 59 = 301, 301 - 59 = 242, 242 / 2 = 121. So option A is 121. Wait, but that seems large. Wait, maybe the diagram is such that the angle at C is actually the angle between tangent and chord, but no, the angle at C is not at A. Wait, maybe the problem has a typo, but according to the formula, it's 121. Wait, but the options include 121 as A. So I think the correct answer is A. 121.

Wait, no, wait, I think I made a mistake. The angle formed by a tangent and a secant is equal to half the measure of the intercepted arc that is "outside" the angle. Wait, no, the correct theorem is: the measure of an angle formed by a tangent and a secant drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc and the minor arc between the two points of intersection. So in this case, the tangent is AC (intersecting at A) and the secant is CB (intersecting at B and... Wait, in the diagram, B is on the circle, so the secant is CB, intersecting the circle at B and... Wait, maybe the diagram has only B as the other intersection, meaning it's a tangent and a chord (but CB is a chord, not a secant). Wait, no, a secant intersects the circle at two points, so if CB intersects at B and another point, but in the diagram, it's drawn as CB intersecting at B, so maybe it's a tangent and a chord (AB). Then the angle at C would be equal to 90 degrees minus the angle at A. Wait, no, this is confusing. Let's go back to the tangent-radius: \( OA \perp AC \), so \( \angle OAC = 90^\circ \). The central angle \( \angle AOB = 59^\circ \), so the inscribed angle over arc AB is \( 29.5^\circ \), but that's not helpful. Wait, maybe the answer is 31, as 90 - 59 = 31? Wait, 90 - 59 = 31, which is option C. Ah! That makes sense. Because \( \angle OAC = 90^\circ \), and \( \angle AOB = 59^\circ \), so in triangle OAC, but wait, no, triangle OBC? No, wait, the angle at C is equal to 90 degrees minus half the central angle? No, 90 - 59 = 31, which is option C. So that's the answer.

Answer:

C. 31