Given that 𝐴𝐵 is a diameter in circle 𝑀 and the measure of angle 𝐷𝑀𝐵 is equal to five 𝑥 plus 12 degrees, determine the measure of 𝐴𝐶.
Now, firstly, let’s clarify what’s meant by some of the notation in this question — specifically this part here 𝐴𝐶 with a circumflex over the top. This refers to the portion of the circle connecting the points 𝐴 and 𝐶, which is known as an arc. The measure of an arc is defined to be its central angle. So in this case, that’s four 𝑥 degrees. In order to calculate the measure of the arc 𝐴𝐶, we need to know what the value of 𝑥 is.
Let’s look at the other information given in the question. We’re told that the measure of the angle 𝐷𝑀𝐵 is five 𝑥 plus 12 degrees. So I’ve added that expression to the diagram. The other key piece of information in the question is that the line 𝐴𝐵 is a diameter of the circle 𝑀.
Remember the diameter of a circle is a line whose endpoints are both on the circumference of a circle and it must pass through the centre of the circle. A diameter divides a circle up into two semicircles. The measure of a semicircle is 180 degrees as its central angle is a straight line. And the sum of angles on a straight line is 180 degrees.
In this case, the central angle of the arc 𝐴𝐵 is the sum of the angle of five 𝑥 plus 12 degrees and the angle of two 𝑥 degrees. So we have the equation five 𝑥 plus 12 plus two 𝑥 is equal to 180. We can solve this equation to find the value of 𝑥. Simplifying the left-hand side of the equation gives seven 𝑥 plus 12 is equal to 180. Subtracting 12 from each side gives seven 𝑥 is equal to 168. The final step in solving this equation is to divide both sides by seven, which gives 𝑥 is equal to 24.
So we found the value of 𝑥. Remember the reason we wanted to do this is so we could calculate the measure of the arc 𝐴𝐶, which is equal to four 𝑥 degrees. The measure of the arc 𝐴𝐶 is equal to four multiplied by 24. It’s 96 degrees.
Remember there were two key facts that we used within this question. Firstly, the measure of an arc is defined as being equal to its central angle. Secondly, a special case of this, the measure of a semicircle is 180 degrees.