Question Video: Finding an Arc’s Measure in a Circle Given the Other Arcs’ Measures by Solving Linear Equations | Nagwa Question Video: Finding an Arc’s Measure in a Circle Given the Other Arcs’ Measures by Solving Linear Equations | Nagwa

Question Video: Finding an Arc’s Measure in a Circle Given the Other Arcs’ Measures by Solving Linear Equations Mathematics • Third Year of Preparatory School

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Given that the line segment 𝐴𝐵 is a diameter in a circle of center 𝑀 and m of the arc 𝐴𝐶 : m of the arc 𝐷𝐵 = 85 : 67, determine m of the arc 𝐴𝐶𝐷.

05:07

Video Transcript

Given that the line segment 𝐴𝐵 is a diameter in a circle of center 𝑀 and the ratio of the measure of the arc 𝐴𝐶 to the measure of the arc 𝐷𝐵 is 85 to 67, determine the measure of the arc 𝐴𝐶𝐷.

We recall first that this notation here means the measure of the minor arc between points 𝐴 and 𝐶. That’s this arc here on the diagram. The measure of the arc 𝐷𝐵 means the measure of the minor arc between the points 𝐷 and 𝐵. That’s this arc here. We’re given in the question that the ratio of the measure of the arc 𝐴𝐶 to the measure of the arc 𝐷𝐵 is 85 to 67. We want to determine the measure of the arc 𝐴𝐶𝐷, the minor arc from 𝐴 to 𝐶 to 𝐷. This arc consists of two adjacent arcs, 𝐴𝐶 and 𝐶𝐷. And so, the measure of the arc 𝐴𝐶𝐷 is the sum of the measures of these two arcs. It’s the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐶𝐷.

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 𝐶𝐷 is 28 degrees because this is the measure of its central angle. The measure of the arc 𝐴𝐶 will be the measure of the central angle 𝐴𝑀𝐶, which is unknown. And the measure of the arc 𝐷𝐵 will be the measure of the central angle 𝐵𝑀𝐷, which is also unknown. However, as the line segment 𝐴𝐵 is a diameter of the circle, the sum of these three angles, and hence the sum of the measures of the arcs, 𝐴𝐶, 𝐶𝐷, and 𝐷𝐵, is 180 degrees because the line segment 𝐴𝐵 is a straight line.

We therefore have an equation. The measure of the arc 𝐷𝐵 plus 28 degrees plus the measure of the arc 𝐴𝐶 equals 180 degrees. We can simplify by subtracting 28 degrees from each side. And we find that the measure of the arc 𝐷𝐵 plus the measure of the arc 𝐴𝐶 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 𝐴𝐶 divided by the measure of the arc 𝐷𝐵 is equal to 85 over 67. Cross multiplying gives 67 times the measure of the arc 𝐴𝐶 is equal to 85 times the measure of the arc 𝐷𝐵. We can then divide both sides of this equation by 85 to find that the measure of the arc 𝐷𝐵 is equal to 67 times the measure of the arc 𝐴𝐶 over 85.

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

And by multiplying both sides of the equation by 85, it follows that the measure of the arc 𝐴𝐶 is 85 degrees. We now know the measure of the arc 𝐴𝐶 and the measure of the arc 𝐶𝐷. So we can find the measure of the arc 𝐴𝐶𝐷 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 𝐷𝐵 in terms of the measure of the arc 𝐴𝐶. 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 𝐴𝐶 was 85 degrees and the measure of the arc 𝐷𝐵 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 𝐴𝐶𝐷 is 113 degrees.

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