Find the value of 𝑥 given that the measure of the arc 𝐴𝐷 equals 34 degrees.
Let’s look carefully at the diagram we’ve been given. There is a circle with its center at the point 𝑀. Points 𝐴 and 𝐵 lie on the circumference of the circle. And the line segment connecting them passes through the center, so it is a diameter. There are two further points on the circle’s circumference, point 𝐶 and point 𝐷. And the radii connecting each of these points to the center of the circle, the line segments 𝐶𝑀 and 𝐷𝑀, have also been drawn.
We’re given in the question that the measure of the arc 𝐴𝐷 is 34 degrees. We should recall that this notation means the minor arc connecting points 𝐴 and 𝐷. It’s this arc here now highlighted in pink. We are asked to determine the value of 𝑥, which we can see is the measure of the angle formed between the two radii 𝐷𝑀 and 𝐴𝑀.
Another way of describing this angle is as the central angle of the arc 𝐴𝐷. It’s the angle formed between the two radii connecting the endpoints of the arc 𝐴𝐷 to the center of the circle. We can therefore recall a definition which relates the measure of an arc to the measure of its central angle. The measure of an arc is equal to the measure of its central angle. So, we have that the measure of the arc 𝐴𝐷 is equal to 𝑥 degrees.
The measure of the arc 𝐴𝐷 is 34 degrees, so we have 34 degrees equals 𝑥 degrees. And as both sides of this equation are measured in degrees, we find that 𝑥 is equal to 34. So by recalling that the measure of an arc is the measure of its central angle, we found the value of 𝑥 is 34.