# Video: Finding the Measure of an Arc in a Circle given a Diameter and the Measures of Two Central Angles in the Form of Algebraic Expression

Given that π΄π΅ is a diameter in circle π and πβ π·ππ΅ = (5π₯ + 12)Β°, determine π π΄πΆ.

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### Video Transcript

Given that π΄π΅ is a diameter in circle π and the measure of angle π·ππ΅ is equal to five π₯ plus 12 degrees, determine the measure of π΄πΆ.

Now, firstly, letβs clarify whatβs meant by some of the notation in this question β specifically this part here π΄πΆ with a circumflex over the top. This refers to the portion of the circle connecting the points π΄ and πΆ, which is known as an arc. The measure of an arc is defined to be its central angle. So in this case, thatβs four π₯ degrees. In order to calculate the measure of the arc π΄πΆ, we need to know what the value of π₯ is.

Letβs look at the other information given in the question. Weβre told that the measure of the angle π·ππ΅ is five π₯ plus 12 degrees. So Iβve added that expression to the diagram. The other key piece of information in the question is that the line π΄π΅ is a diameter of the circle π.

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 π΄π΅ is the sum of the angle of five π₯ plus 12 degrees and the angle of two π₯ degrees. So we have the equation five π₯ plus 12 plus two π₯ is equal to 180. We can solve this equation to find the value of π₯. Simplifying the left-hand side of the equation gives seven π₯ plus 12 is equal to 180. Subtracting 12 from each side gives seven π₯ is equal to 168. The final step in solving this equation is to divide both sides by seven, which gives π₯ is equal to 24.

So we found the value of π₯. Remember the reason we wanted to do this is so we could calculate the measure of the arc π΄πΆ, which is equal to four π₯ degrees. The measure of the arc π΄πΆ is equal to four multiplied by 24. Itβ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.