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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

Given that 𝑀 is the center of the circle, find π‘šβˆ π‘€π‘‹π‘Œ.

05:45

Video Transcript

Given that 𝑀 is the center of the circle, find the measure of angle π‘€π‘‹π‘Œ.

We will begin by finding angle π‘€π‘‹π‘Œ 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 π‘€π‘‹π‘Œ in orange. Let’s now collect any pertinent information that is given in the diagram of the circle. We see that angle π‘‹π‘€π‘Œ is given a measure of 102 degrees. That is the only angle measure that is given on this diagram. We also notice that chord 𝐴𝐡 has an equal length to chord 𝐴𝐢. Since 𝐴𝑋 equals 𝑋𝐡, we identify 𝑋 as the midpoint of chord 𝐴𝐡. And given that πΆπ‘Œ equals π‘Œπ΄, we also conclude that π‘Œ is the midpoint of chord 𝐴𝐢.

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 𝐴𝐡 and 𝐴𝐢 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 π‘Œ are the midpoints of the two chords and 𝑀 is the center of the circle, the line segments 𝑀𝑋 and π‘€π‘Œ must be the perpendicular bisectors of the two chords. This tells us that 𝑀𝑋 is actually the distance from chord 𝐴𝐡 to the center of the circle and that π‘€π‘Œ is the distance from chord 𝐴𝐢 to the center.

Since the two chords 𝐴𝐡 and 𝐴𝐢 have equal lengths, they must also be equidistant from the center. And this tells us that 𝑀𝑋 equals π‘€π‘Œ. Having marked 𝑀𝑋 equal to π‘€π‘Œ in pink on the diagram, we will now clear some space.

Now, we turn our attention to triangle π‘€π‘‹π‘Œ. Since two sides of this triangle have equal lengths, triangle π‘€π‘‹π‘Œ 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 π‘€π‘Œ means that the measure of angle π‘€π‘‹π‘Œ equals the measure of angle π‘€π‘Œπ‘‹. This fact must be important given that we are looking for the measure of angle π‘€π‘‹π‘Œ. So we mark these two angles as congruent on the diagram.

Now we know quite a lot about the three interior angles of triangle π‘€π‘‹π‘Œ. We know that the measure of angle π‘‹π‘€π‘Œ 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 π‘€π‘‹π‘Œ, this means that the measure of angle π‘‹π‘€π‘Œ plus the measure of angle π‘€π‘‹π‘Œ plus the measure of angle π‘€π‘Œπ‘‹ equals 180 degrees. We know that the measure of angle π‘‹π‘€π‘Œ equals 102 degrees and also that the measure of angle π‘€π‘‹π‘Œ equals the measure of angle π‘€π‘Œπ‘‹. So we will substitute these expressions into the equation. This gives us 102 degrees plus the measure of angle π‘€π‘‹π‘Œ plus the measure of angle π‘€π‘‹π‘Œ equals 180 degrees.

Since the measure of angle π‘€π‘‹π‘Œ is added to itself, we can simplify this as two times the measure of angle π‘€π‘‹π‘Œ. Then subtracting 102 degrees from each side of the equation leads us to two times the measure of angle π‘€π‘‹π‘Œ equals 78 degrees. Finally, after dividing both sides of the equation by two, we have our answer. The measure of angle π‘€π‘‹π‘Œ 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 π‘€π‘‹π‘Œ is isosceles and verifying the sum of the interior angles is 180 degrees, we are confident in our final answer.

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