### Video Transcript

Given that the line segment π΄π΅ is a diameter in a circle of center π and the ratio of the measure of the arc π΄πΆ to the measure of the arc π·π΅ is 85 to 67, determine the measure of the arc π΄πΆπ·.

We recall first that this notation here means the measure of the minor arc between points π΄ and πΆ. Thatβs this arc here on the diagram. The measure of the arc π·π΅ means the measure of the minor arc between the points π· and π΅. Thatβs this arc here. Weβre given in the question that the ratio of the measure of the arc π΄πΆ to the measure of the arc π·π΅ is 85 to 67. We want to determine the measure of the arc π΄πΆπ·, the minor arc from π΄ to πΆ to π·. This arc consists of two adjacent arcs, π΄πΆ and πΆπ·. And so, the measure of the arc π΄πΆπ· is the sum of the measures of these two arcs. Itβs the measure of the arc π΄πΆ plus the measure of the arc πΆπ·.

Next, we recall that the measure of an arc is equal to the measure of its central angle. That is the angle formed by the two radii connecting the endpoints of the arc to the center of the circle. We can therefore deduce that the measure of the arc πΆπ· is 28 degrees because this is the measure of its central angle. The measure of the arc π΄πΆ will be the measure of the central angle π΄ππΆ, which is unknown. And the measure of the arc π·π΅ will be the measure of the central angle π΅ππ·, which is also unknown. However, as the line segment π΄π΅ is a diameter of the circle, the sum of these three angles, and hence the sum of the measures of the arcs, π΄πΆ, πΆπ·, and π·π΅, is 180 degrees because the line segment π΄π΅ is a straight line.

We therefore have an equation. The measure of the arc π·π΅ plus 28 degrees plus the measure of the arc π΄πΆ equals 180 degrees. We can simplify by subtracting 28 degrees from each side. And we find that the measure of the arc π·π΅ plus the measure of the arc π΄πΆ is 152 degrees. Now, we know that the measures of these two arcs are in the ratio 85 to 67. In other words, the measure of the arc π΄πΆ divided by the measure of the arc π·π΅ is equal to 85 over 67. Cross multiplying gives 67 times the measure of the arc π΄πΆ is equal to 85 times the measure of the arc π·π΅. We can then divide both sides of this equation by 85 to find that the measure of the arc π·π΅ is equal to 67 times the measure of the arc π΄πΆ over 85.

We can now take this expression for the measure of the arc π·π΅ in terms of the measure of the arc π΄πΆ and substitute it into our earlier equation, which will give an equation in terms of the measure of the arc π΄πΆ only. Doing so gives 67 times the measure of the arc π΄πΆ over 85 plus the measure of the arc π΄πΆ equals 152 degrees. The second term can be thought of as 85 over 85 times the measure of the arc π΄πΆ. So summing the two terms on the left-hand side gives 152 times the measure of the arc π΄πΆ over 85. We can then cancel a factor of 152 from each side of the equation, giving the measure of the arc π΄πΆ over 85 equals one degree.

And by multiplying both sides of the equation by 85, it follows that the measure of the arc π΄πΆ is 85 degrees. We now know the measure of the arc π΄πΆ and the measure of the arc πΆπ·. So we can find the measure of the arc π΄πΆπ· by finding their sum, 85 degrees plus 28 degrees, which is 113 degrees.

Now, we may not actually have needed to work through all of the formal method for finding an expression for the measure of the arc π·π΅ in terms of the measure of the arc π΄πΆ. If we had observed that the sum of the two parts of the ratio 85 and 67 is in fact equal to 152, which was the remaining arc measure, we could see straightaway that the measure of the arc π΄πΆ was 85 degrees and the measure of the arc π·π΅ was 67 degrees. And these two values were in the correct ratio. This would have speeded up our working a little, but it would have led to the same answer, which is that the measure of the arc π΄πΆπ· is 113 degrees.