Question Video: Finding the Measure of an Angle Using the Properties of Tangents to the Circle Mathematics • 11th Grade

Video Transcript

Given that 𝐴𝐷 is a tangent to the circle and the measure of angle 𝐷𝐴𝐶 is 90 degrees, calculate the measure of angle 𝐴𝐶𝐵.

We are told in the question that 𝐴𝐷 is a tangent and the measure of angle 𝐷𝐴𝐶 is 90 degrees. We know that the tangent to any circle is perpendicular to the radius or diameter. This means that, in this question, 𝐴𝐶 is a diameter of the circle. We could use two possible angle properties or circle theorems to solve this problem. Firstly, we could use the fact that the angle in a semicircle equals 90 degrees. This means that the measure of angle 𝐴𝐵𝐶 is 90 degrees. We could also have found this using the alternate segment theorem, where the measure of angle 𝐷𝐴𝐶 is equal to the measure of angle 𝐴𝐵𝐶. Either way, we know that 𝐴𝐵𝐶 is equal to 90 degrees.

We now need to solve the equation nine 𝑥 is equal to 90. Dividing both sides of this equation by nine gives us 𝑥 is equal to 10. Our value of 𝑥 is 10 degrees. We can see on the diagram that the measure of angle 𝐴𝐶𝐵 is five 𝑥. As 𝑥 is equal to 10 degrees, we need to multiply this by five. Five multiplied by 10 is equal to 50. Therefore, angle 𝐴𝐶𝐵 equals 50 degrees.