Question Video: Finding the Measure of an Arc Using the Relationship Between the Sides of an Inscribed Rectangle Mathematics

In the following figure, a rectangle 𝐴𝐡𝐢𝐷 is inscribed in a circle, where the measure of arc 𝐴𝐡 = 71°. Find the measure of arc 𝐴𝐷.

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Video Transcript

In the following figure, a rectangle 𝐴𝐡𝐢𝐷 is inscribed in a circle, where the measure of arc 𝐴𝐡 equals 71 degrees. Find the measure of arc 𝐴𝐷.

We’re going to use the fact that 𝐴𝐡𝐢𝐷 is a rectangle. This means that the line segment or chord 𝐴𝐡 is parallel to chord 𝐷𝐢. Similarly, line segment 𝐷𝐴 must be parallel to line segment 𝐡𝐢. This means we can use the theorem that tells us that the measure of arcs between parallel chords of a circle are equal. Since line segments 𝐴𝐡 and 𝐷𝐢 are parallel, the measure of arc 𝐴𝐷 must be equal to the measure of arc 𝐡𝐢. Similarly, the measure of arc 𝐴𝐡 must be equal to the measure of arc 𝐷𝐢. But we’re actually told that’s 71 degrees.

Now, since the sum of all the arc measures that make up the circle is 360 degrees, we can form and solve an equation. We know that the measure of arc 𝐴𝐡 and the measure of arc 𝐷𝐢 is 71. So our equation is the measure of arc 𝐴𝐷 plus the measure of arc 𝐡𝐢 plus 71 plus 71 equals 360. Since 𝐴𝐷 and 𝐡𝐢 are congruent arcs, we can further simplify this. Two times the measure of arc 𝐴𝐷 plus 142 degrees equals 360 degrees. Then, we subtract 142 degrees from both sides, and our final stage is to divide through by two. So the measure of arc 𝐴𝐷 is 218 divided by two, which is equal to 109 or 109 degrees. The measure of arc 𝐴𝐷 is 109 degrees.

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