Question Video: Using Theories of Parallel Chords and Tangent Relationships to Find the Measure of an Arc Mathematics

In the following figure, 𝑀 is a circle. Segment 𝐴𝐡 and segment 𝐢𝐷 are two chords of the circle, and line 𝐸𝐹 is a tangent to the circle at 𝐸. If segment 𝐴𝐡 βˆ₯ segment 𝐢𝐷 βˆ₯ 𝐸𝐹, the measure of arc 𝐴𝐢 = 30Β°, and the measure of arc 𝐷𝐸 = 74Β°, find the measure of arc 𝐴𝐡.

01:54

Video Transcript

In the following figure, 𝑀 is a circle. Segment 𝐴𝐡 and segment 𝐢𝐷 are two chords of the circle, and line 𝐸𝐹 is a tangent to the circle at 𝐸. If segment 𝐴𝐡 is parallel to segment 𝐢𝐷 which is parallel to line 𝐸𝐹, the measure of arc 𝐴𝐢 equals 30 degrees, and the measure of arc 𝐷𝐸 equals 74 degrees, find the measure of arc 𝐴𝐡.

Let’s add some information we’re given to the figure. Arc 𝐴𝐢 measures 30 degrees. Arc 𝐷𝐸 measures 74 degrees. The key point here is that we’re working with parallel chords and a parallel tangent line. The measures of the arcs between parallel chords in a circle are equal. Therefore, arc 𝐡𝐷 will be equal in measure to arc 𝐴𝐢. And similarly, the measures of the arcs between a parallel chord and a tangent of that circle are equal. In our circle, that means that arc 𝐷𝐸 will be equal in measure to arc 𝐢𝐸. We can therefore add the measure of arc 𝐡𝐷 equals 30 degrees and the measure of arc 𝐢𝐸 equals 74 degrees onto our figure.

Our missing arc is 𝐴𝐡. We know that the sum of the measures of all the arcs in a circle must equal 360 degrees. Therefore, the measure of arc 𝐴𝐡 will be equal to 360 degrees minus the sum of all the other four arcs we know: 74 degrees plus 74 degrees plus 30 degrees plus 30 degrees. 360 degrees minus 208 degrees equals 152 degrees. The measure of arc 𝐴𝐡 in this figure is 152 degrees.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.