Given that line segment 𝐴𝐵 is a tangent to the circle 𝑀 at the point 𝐵, the measure of angle 𝐵𝑀𝐶 is equal to three times the measure of angle 𝐴, and the point 𝐶 is the midpoint of line segment 𝐷𝐸, find the value of 𝑥.
First of all, let’s list out the information we were given. We know that 𝐴𝐵 is tangent to the circle at point 𝐵. We know that the measure of angle 𝐵𝑀𝐶 is three times the measure of angle 𝐴. 𝐶 is the midpoint of 𝐷𝐸. And by looking at our figure, we see one other angle measure. We see that the measure of angle 𝐵𝑀𝐶 is equal to three 𝑥. From these properties, let’s see if we can draw any conclusions. Because 𝐴𝐵 is tangent to this circle at point 𝐵 and point 𝑀 is the center of the circle, the radius 𝑀𝐵 will be creating a right angle with the line segment 𝐴𝐵. The radius is perpendicular to the tangent at point 𝐵. We can then say that the measure of angle 𝐴𝐵𝑀 is 90 degrees.
After that, because 𝐶 is the midpoint of line segment 𝐷𝐸, we’re showing that the line from the center of the circle 𝑀 to point 𝐶 bisects the chord 𝐸𝐷. And when that’s the case, it is a perpendicular bisector. There will be a right angle here. So we can say that the measure of angle 𝑀𝐶𝐴 is 90 degrees. We can also say that 𝐴𝐵𝑀𝐶 is a quadrilateral, which means the interior angles must sum to 360 degrees. At this point, it seems like we know something about three of the four angles inside this quadrilateral. But then we remember that the measure of angle 𝐵𝑀𝐶 is equal to three times the measure of angle 𝐴. And the measure of angle 𝐵𝑀𝐶 is three 𝑥 degrees.
If we plug in three 𝑥 degrees for the measure of angle 𝐵𝑀𝐶, we get the equation three 𝑥 degrees is equal to three times the measure of angle 𝐴. By dividing both sides of this equation by three, we can see that the measure of angle 𝐴 will be equal to 𝑥 degrees. And so we’ll add that to our graph so that we can create the equation 90 degrees plus three 𝑥 degrees plus 90 degrees plus 𝑥 degrees will equal 360 degrees. We’ll combine like terms so that we have 180 degrees plus four 𝑥 degrees equals 360 degrees. If we subtract 180 degrees from both sides, we find out that four 𝑥 degrees equals 180 degrees. From there, we’ll divide by four on both sides of the equation to get 𝑥 by itself. And we’ll see that 𝑥 degrees equals 45 degrees. Therefore, 𝑥 equals 45.