Question Video: Finding the Measure of an Angle in a Triangle inside a Circle Where Two of Its Vertices Intersect with Chords and Its Third Is the Circle’s Center Mathematics

Video Transcript

Given that 𝑀 is the center of the circle, find the measure of angle 𝑀𝑋𝑌.

We will begin by finding angle 𝑀𝑋𝑌 on the diagram. We recall that when an angle is named with three letters, the middle letter is the vertex of the angle. So in this case, our angle has vertex 𝑋. In the given diagram of the circle with center 𝑀, we have highlighted angle 𝑀𝑋𝑌 in orange. Let’s now collect any pertinent information that is given in the diagram of the circle. We see that angle 𝑋𝑀𝑌 is given a measure of 102 degrees. That is the only angle measure that is given on this diagram. We also notice that chord 𝐴𝐵 has an equal length to chord 𝐴𝐶. Since 𝐴𝑋 equals 𝑋𝐵, we identify 𝑋 as the midpoint of chord 𝐴𝐵. And given that 𝐶𝑌 equals 𝑌𝐴, we also conclude that 𝑌 is the midpoint of chord 𝐴𝐶.

When we consider that we have two chords of equal length in the given diagram, this reminds us of a theorem that says that two chords of equal lengths in the same circle are equidistant from the center of the circle. We also know that the distance of a chord from the center of the circle is measured by the length of the segment from the center that perpendicularly bisects the chord.

In this example, we have two chords 𝐴𝐵 and 𝐴𝐶 that have equal lengths. And we recall that the perpendicular bisector of a chord will always go through the center of the circle. Since 𝑋 and 𝑌 are the midpoints of the two chords and 𝑀 is the center of the circle, the line segments 𝑀𝑋 and 𝑀𝑌 must be the perpendicular bisectors of the two chords. This tells us that 𝑀𝑋 is actually the distance from chord 𝐴𝐵 to the center of the circle and that 𝑀𝑌 is the distance from chord 𝐴𝐶 to the center.

Since the two chords 𝐴𝐵 and 𝐴𝐶 have equal lengths, they must also be equidistant from the center. And this tells us that 𝑀𝑋 equals 𝑀𝑌. Having marked 𝑀𝑋 equal to 𝑀𝑌 in pink on the diagram, we will now clear some space.

Now, we turn our attention to triangle 𝑀𝑋𝑌. Since two sides of this triangle have equal lengths, triangle 𝑀𝑋𝑌 is considered an isosceles triangle. We recall the isosceles triangle theorem, which says if two sides of a triangle are congruent, then the angles opposite those sides are congruent. Knowing that congruent means equal in measure and having 𝑀𝑋 equal to 𝑀𝑌 means that the measure of angle 𝑀𝑋𝑌 equals the measure of angle 𝑀𝑌𝑋. This fact must be important given that we are looking for the measure of angle 𝑀𝑋𝑌. So we mark these two angles as congruent on the diagram.

Now we know quite a lot about the three interior angles of triangle 𝑀𝑋𝑌. We know that the measure of angle 𝑋𝑀𝑌 is 102 degrees and that the measure of the other two angles are equal. We also recall that the interior angles in a triangle sum to 180 degrees. In the case of triangle 𝑀𝑋𝑌, this means that the measure of angle 𝑋𝑀𝑌 plus the measure of angle 𝑀𝑋𝑌 plus the measure of angle 𝑀𝑌𝑋 equals 180 degrees. We know that the measure of angle 𝑋𝑀𝑌 equals 102 degrees and also that the measure of angle 𝑀𝑋𝑌 equals the measure of angle 𝑀𝑌𝑋. So we will substitute these expressions into the equation. This gives us 102 degrees plus the measure of angle 𝑀𝑋𝑌 plus the measure of angle 𝑀𝑋𝑌 equals 180 degrees.

Since the measure of angle 𝑀𝑋𝑌 is added to itself, we can simplify this as two times the measure of angle 𝑀𝑋𝑌. Then subtracting 102 degrees from each side of the equation leads us to two times the measure of angle 𝑀𝑋𝑌 equals 78 degrees. Finally, after dividing both sides of the equation by two, we have our answer. The measure of angle 𝑀𝑋𝑌 equals 39 degrees.

It’s a good idea at the end to do a quick computational check. We want to see if the three interior angles 102 degrees, 39 degrees, and 39 degrees add up to the sum that we expected. Having shown that triangle 𝑀𝑋𝑌 is isosceles and verifying the sum of the interior angles is 180 degrees, we are confident in our final answer.