In the following figure, line segment 𝐵𝐶 is a diameter of the circle. The measure of angle a 𝐴𝐶𝐵 is equal to the measure of angle 𝐶𝐷𝐸. The measure of arc 𝐵𝐷 is 42 degrees. And 𝐸 is the midpoint of arc 𝐶𝐷. Find the measure of arc 𝐴𝐶.
There is a lot of information here. And before starting, it is worth recalling what we mean by an inscribed angle. An inscribed angle is formed by the intersection of two chords on the circumference
of a circle. For example, the chords 𝐴𝐶 and 𝐵𝐶 intersect at point 𝐶. This means that angle 𝐴𝐶𝐵 is subtended by the arc 𝐴𝐵. And we conclude that the measure of the angle is equal to the measure of the arc.
We are told that the Angles, 𝐴𝐶𝐵 and 𝐶𝐷𝐸 are equal. This means that the arcs 𝐴𝐵 and 𝐶𝐸 are also equal. We are told that the measure of arc 𝐵𝐷 is 42 degrees. Since 𝐸 is the midpoint of Arc 𝐶𝐷, then the triangle 𝐶𝐷𝐸 is isosceles, where
𝐶𝐸 is equal to 𝐷𝐸. We know that two angles in an isosceles triangle are equal. And this also means that the arc 𝐷𝐸 is equal to the arc 𝐶𝐸 and the arc 𝐴𝐵.
Our aim in this question is to find the measure of arc 𝐴𝐶. Since 𝐵𝐶 is a diameter, we know it splits the circle in half. Since the angles in a circle sum to 360 degrees, in a semicircle, they must sum to
180 degrees. The measure of arcs 𝐵𝐷, 𝐷𝐸, and 𝐶𝐸 must sum to 180 degrees. This gives us an equation 42 degrees plus 𝑥 plus 𝑥 is equal to 180 degrees. 𝑥 plus 𝑥 is equal to two 𝑥. And subtracting 42 degrees from both sides, this is equal to 138 degrees. We can then divide through by two such that 𝑥 is equal to 69 degrees. This means that the measure of the arcs 𝐷𝐸, 𝐶𝐸, and 𝐴𝐵 are all equal to 69
Since all five arcs on the circumference of our circle sum to 360 degrees, we can use
this to calculate the measure of arc 𝐴𝐶. Adding three lots of 69 to 42 gives us 249. The measure of our arc a𝐶 plus 249 degrees is equal to 360 degrees. We can then subtract 249 degrees from both sides such that the measure of arc 𝐴𝐶 is
equal to 111 degrees.