Video: Finding the Measure of an Arc in a Circle given the Other Arcs’ Measures

Determine the measure of arc 𝐢𝐡.

03:08

Video Transcript

Determine the measure of arc 𝐢𝐡.

First, let’s see what we know. We know that the line segment 𝐢𝐴 and the line segment 𝐸𝐴 intersect outside the circle at point 𝐴. Both line segment 𝐢𝐴 and line segment 𝐸𝐴 are secants of the circle. The figure has also told us that the measure of arc 𝐢𝐡 is equal to the measure of arc 𝐸𝐷. We can also say that the sum of all arc measures in the circle must be 360 degrees. We’re interested in finding the measure of arc 𝐢𝐡. But in order to do that, we’ll need to find the measure of arc 𝐡𝐷.

Based on what we said initially about the intersection outside the circle, we can say that the angle made by two intersecting lines outside the circle is half the positive difference of the intercepted arcs, which is the intersecting secant angles theorem. The angle created by these intersecting lines in our case is 34 degrees. 34 degrees is equal to the measure of the intercepted arc 𝐢𝐸, which is 151 degrees, minus the measure of the intercepted arc 𝐡𝐷 and then divided it by two.

We need to solve for the measure of the arc 𝐡𝐷. So, we multiply both sides of the equation by two, which gives us 68 equals 151 minus the measure of arc 𝐡𝐷. So, we subtract 151 from both sides, and we get negative 83 degrees equals the negative measure of arc 𝐡𝐷. We multiply both sides of the equation by negative one and flip the sides, and we get the measure of arc 𝐡𝐷 equals positive 83 degrees. But this is not our final answer. We’re still trying to find the measure of arc 𝐢𝐡.

But remember, we know that all of these arcs must sum to 360 degrees. And we know that arc 𝐢𝐡 and arc 𝐸𝐷 must be equal. We could say that they’re equal to π‘₯ degrees. If we do that, we could then create the equation 151 plus π‘₯ plus π‘₯ plus 83 equals 360. If we combine like terms, 151 plus 83 equals 234. π‘₯ plus π‘₯ equals two π‘₯. Therefore, 234 plus two π‘₯ equals 360 degrees. We subtract 234 from both sides of the equation, which tells us two π‘₯ equals 126. Divide both sides by two, and we see that π‘₯ equals 63. This tells us that both arc 𝐢𝐡 and arc 𝐸𝐷 is equal to a measure of 63 degrees. We were primarily interested in the measure of arc 𝐢𝐡, and we found that that measure is 63 degrees.

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