Question Video: Using Theories of Parallel Chords to Find the Measure of an Arc Mathematics • Third Year of Preparatory School

Video Transcript

In the following figure, 𝐴𝐵𝐶 is an equilateral triangle inscribed in a circle. Segment 𝐴𝐵 and segment 𝐷𝐸 are two chords of the circle, and line 𝐹𝐺 is a tangent to the circle at 𝐹. If segment 𝐷𝐸 is parallel to segment 𝐴𝐵, which is parallel to line 𝐹𝐺, and the measure of arc 𝐶𝐷 is 79 degrees, find the measure of 𝐸𝐵. Find the measure of 𝐷𝐹.

The first thing we can do is add in the information we know into our circle. 𝐴𝐵𝐶 is an equilateral triangle. Therefore, each of their interior angles measures 60 degrees. We also need to note that 𝐷𝐸 is parallel to 𝐴𝐵, which is parallel to 𝐹𝐺. Because we know that these lines are parallel, we can also say the arcs between them will be equal in measure. This means the measure of arc 𝐷𝐴 will be equal to the measure of arc 𝐸𝐵. And the measure of arc 𝐴𝐹 will be equal to the measure of arc 𝐹𝐵. We know the first two chords will be equal in measure by the measure of arcs between parallel chords theorem and the second set, arc 𝐴𝐹 and arc 𝐹𝐵, by the measure of arcs between a parallel chord and tangent theorem. We know that arc 𝐶𝐷 equals 79 degrees.

The first arc we need to find the measure of is 𝐸𝐵. To do that, let’s consider the inscribed angle 𝐶𝐵𝐴. Using this inscribed angle, we can identify the measure of arc 𝐴𝐶. The arc 𝐴𝐶 is two times the measure of this inscribed angle, which we know is 60 degrees. Therefore, the arc 𝐴𝐶 measures 120 degrees. We know that the larger arc 𝐴𝐶 is made up of the two smaller arcs 𝐴𝐷 and 𝐷𝐶. Therefore, we can say that 120 degrees equals the measure of arc 𝐴𝐷 plus 79 degrees. Subtracting 79 degrees from both sides, we find the measure of arc 𝐴𝐷 to be 41 degrees. And we’ve already shown that the measure of arc 𝐸𝐵 will be equal to the measure of arc 𝐴𝐷 by the measure of arcs between parallel chords theorem. This makes the measure of arc 𝐸𝐵 equal to 41 degrees.

Our next unknown arc measure that we’re looking for is 𝐷𝐹. To do that, let’s first look at the inscribed angle 𝐴𝐶𝐵. The inscribed angle 𝐴𝐶𝐵 subtends the arc 𝐴𝐵. Therefore, we can say the measure of arc 𝐴𝐵 equals two times the measure of angle 𝐴𝐶𝐵, which again will be 120 degrees. Since the measure of arc 𝐴𝐵 equals the measure of arc 𝐴𝐹 plus the measure of arc 𝐹𝐵 and since arc 𝐹𝐵 and arc 𝐴𝐹 will be equal in measure, we can substitute two times the measure of arc 𝐴𝐹 in this equation. 120 degrees equals two times the measure of arc 𝐴𝐹. The measure of arc 𝐴𝐹 equals 60 degrees, which of course means that arc 𝐹𝐵 also measures 60 degrees.

Back to our unknown arc, that’s 𝐷𝐹, the measure of arc 𝐷𝐹 equals the measure of arc 𝐷𝐴 plus the measure of arc 𝐴𝐹, which is 41 degrees plus 60 degrees, 101 degrees.