Question

consider circle e. segment cd is the perpendicular bisector of (overline{ab}). what is the measure of the arc intercepted by (angle c)? 180 degrees 90 degrees 45 degrees 22.5 degrees

Explanation:

Step1: Analyze the circle and segments

CD is the perpendicular bisector of AB, so AB and CD are diameters (since they bisect each other at the center E). ∠C is an inscribed angle? Wait, no, point C is on the circle, A is on the circle, E is the center. Wait, AE and CE are radii, and AB and CD are perpendicular (since CD is perpendicular bisector of AB), so triangle AEC is a right triangle? Wait, no, AB and CD are perpendicular diameters, so the arc intercepted by ∠C: ∠C is an inscribed angle? Wait, ∠C intercepts arc AD? No, wait, ∠C is at point C, between CA and CD? Wait, no, looking at the diagram, AB and CD are perpendicular diameters, so arc AB is 180 degrees, arc AD is 90 degrees? Wait, no, ∠C: the angle at C, formed by CA and CD. Wait, AE is a radius, CE is a radius, and AB ⊥ CD, so ∠AEC is 90 degrees. Then ∠C: in triangle AEC, AE = CE (radii), so it's an isosceles right triangle, so ∠C is 45 degrees? No, wait, the question is the measure of the arc intercepted by ∠C. The intercepted arc for an inscribed angle is twice the angle, but wait, ∠C is an inscribed angle intercepting arc AD? Wait, no, AB and CD are perpendicular diameters, so arc AD is 90 degrees? Wait, no, AB is a diameter, CD is a diameter, perpendicular, so the four arcs (AC, CB, BD, DA) are each 90 degrees? Wait, no, AB and CD are perpendicular diameters, so they divide the circle into four 90 - degree arcs. Wait, ∠C: the angle at C, between CA and CD. The arc intercepted by ∠C would be arc AD? Wait, no, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. Wait, but ∠C: let's see, point C, A, and the arc between A and the other end. Wait, maybe I made a mistake. Wait, AB and CD are perpendicular diameters, so E is the center. So AE = CE, and ∠AEC = 90 degrees (since AB ⊥ CD). So triangle AEC is isosceles right - angled, so ∠C = 45 degrees? No, wait, the question is about the arc intercepted by ∠C. Wait, maybe ∠C is an inscribed angle intercepting arc AD. Wait, arc AD: since AB and CD are perpendicular diameters, arc AD is 90 degrees? No, wait, AB is a diameter, so arc AB is 180 degrees. CD is a diameter perpendicular to AB, so arc AC, CB, BD, DA are each 90 degrees? Wait, no, if AB and CD are perpendicular diameters, then the central angles for arcs AC, CB, BD, DA are each 90 degrees. Now, ∠C: the angle at point C, formed by chords CA and CD. The intercepted arc for ∠C would be arc AD. The measure of an inscribed angle is half the measure of its intercepted arc. Wait, but if arc AD is 90 degrees, then the inscribed angle ∠C would be 45 degrees? No, that doesn't match. Wait, maybe I got the intercepted arc wrong. Wait, another approach: AB and CD are perpendicular diameters, so E is the center. So AE = CE, and ∠AEC = 90°, so triangle AEC is isosceles right - angled, so ∠C = 45°, but the question is about the arc intercepted by ∠C. Wait, no, maybe ∠C is an inscribed angle intercepting arc AB? No, AB is a diameter (180 degrees), but that would make the angle 90 degrees. Wait, no, I think I messed up. Wait, the diagram: AB and CD are perpendicular diameters, so E is the center. So ∠C: the angle at C, between CA and CB? No, the problem says "the arc intercepted by ∠C". Wait, maybe ∠C is an inscribed angle intercepting arc AD. Wait, arc AD: since AB and CD are perpendicular, the central angle for arc AD is 90 degrees, so the inscribed angle ∠C (which intercepts arc AD) would be 45 degrees? No, that's not right. Wait, no, maybe ∠C is a central angle? No, C is on the circle. Wait, maybe the arc intercepted by ∠C is arc AB? No, AB is 180 degrees. Wait, no, let's re - read the problem: "What is the measure of the arc intercepted by ∠C?". Let's recall that in a circle, if two diameters are perpendicular, they form four right angles at the center. Now, ∠C: the angle at point C, formed by chords CA and CD. The intercepted arc for ∠C is arc AD. The measure of an inscribed angle is half the measure of its intercepted arc. Wait, but if AB and CD are perpendicular diameters, then arc AD is 90 degrees (central angle 90 degrees), so the inscribed angle ∠C would be 45 degrees? No, that's not one of the options? Wait, no, the options are 180, 90, 45, 22.5. Wait, maybe I made a mistake in identifying the intercepted arc. Wait, maybe ∠C is intercepting arc AB? No, AB is 180 degrees, but an inscribed angle intercepting a semicircle is 90 degrees. Wait, no, ∠C: if we consider triangle AEC, AE = CE (radii), ∠AEC = 90°, so ∠C = 45°, but the arc intercepted by ∠C: wait, maybe the arc is arc AD, which is 90 degrees? No, that's not. Wait, no, the correct approach: AB and CD are perpendicular diameters, so E is the center. So ∠C: the angle at C, between CA and CD. The arc intercepted by ∠C is arc AD. The central angle for arc AD is 90 degrees (since AB ⊥ CD), so the inscribed angle ∠C is half of that, 45 degrees? But the options have 90 degrees. Wait, maybe I got the angle wrong. Wait, maybe ∠C is a central angle? No, C is on the circle. Wait, maybe the arc intercepted by ∠C is arc AB? No, AB is 180 degrees. Wait, no, the diagram: AB and CD are perpendicular diameters, so the arc from A to B is 180 degrees, but ∠C is at point C. Wait, maybe ∠C is intercepting arc AD, but the central angle is 90 degrees, so the arc measure is 90 degrees? No, the inscribed angle is half the arc. Wait, I think I made a mistake. Let's start over. AB and CD are perpendicular bisectors, so they are diameters (since they bisect each other at E, the center). So AB ⊥ CD, so ∠AEC = 90°. Now, ∠C is an inscribed angle? Wait, no, point C, A, and E: AE and CE are radii, so triangle AEC is isosceles right - angled, so ∠C = 45°, but the question is about the arc intercepted by ∠C. Wait, the arc intercepted by ∠C is arc AD. The measure of arc AD: since AB and CD are perpendicular diameters, the central angle for arc AD is 90°, so the arc measure is 90°? No, the arc measure is equal to the central angle. Wait, no, the inscribed angle intercepts an arc, and the arc's measure is twice the inscribed angle. Wait, if ∠C is 45°, then the arc is 90°. Ah! That's it. So the inscribed angle ∠C intercepts arc AD, and the measure of arc AD is 2×∠C. But since AB and CD are perpendicular diameters, arc AD has a central angle of 90°, so the arc measure is 90 degrees. Wait, yes, because the central angle for arc AD is 90° (since AB ⊥ CD), so the arc AD measures 90 degrees, and ∠C, being an inscribed angle intercepting arc AD, would be 45 degrees, but the question is about the arc measure, not the angle measure. So the arc intercepted by ∠C is arc AD, which has a measure of 90 degrees.

Answer:

90 degrees