In the diagram below, triangle ABC is inscribed in the circle and AC=AB. The measure of angle BAC is 42 degrees and segment ED is tangent to the circle at point C. What is the measure of angle ACD?

Atlas Dec 28, 2023

#1**+2 **

Recognize the key property of an inscribed angle: An inscribed angle is half the measure of the central angle that subtends the same arc.

Apply the property to angle BAC: Since angle BAC is inscribed in the circle and subtends arc BC, the measure of central angle BOC is twice the measure of angle BAC. Therefore, BOC = 2 * 42° = 84°.

Identify the tangent-radius relationship: A tangent to a circle from an external point creates a right angle with the radius drawn to the point of contact.

Apply the tangent-radius relationship to angle ACD: Angle DCE is a right angle because DE is tangent to the circle at C. Angle DCB is half of central angle BOC because D is the midpoint of arc BC.

Therefore, in triangle DCB, angles DCB and DCE are complementary, and angle ACD = 180° - (DCB + DCE) = 180° - (84° + 90°) = 56°.

Therefore, the measure of angle ACD is 56 degrees.

BuiIderBoi Dec 28, 2023