Given that the measure of angle 𝑀𝐶𝐵 is equal to 49 degrees, where line segments 𝐴𝐵 and 𝐴𝐶 are tangent to the circle at 𝐵 and 𝐶, find the measure of angle 𝐵𝐴𝑀.
We are told in the question that the measure of angle 𝑀𝐶𝐵 is 49 degrees. And we are asked to find the measure of angle 𝐵𝐴𝑀. We are also told that the line segments 𝐴𝐵 and 𝐴𝐶 are tangent to the circle. We will answer this question by firstly recalling some of the properties of circles known as the circle theorems.
Firstly, we recall that two tangents to a circle that meet at a point are equal in length. This means that the two tangents 𝐴𝐵 and 𝐴𝐶 are equal in length. Next, we recall that a tangent to a circle is perpendicular to the radius at the point of contact. This means that the measures of angle 𝑀𝐵𝐴 and 𝑀𝐶𝐴 are 90 degrees. Since 𝑀𝐵 and 𝑀𝐶 are radii to the circle, they must be equal in length. This means that triangle 𝑀𝐵𝐶 is isosceles. And since two angles in an isosceles triangle are equal, the measure of angle 𝑀𝐵𝐶 is 49 degrees. The measure of angle 𝐴𝐵𝐶 is therefore equal to 90 degrees minus 49 degrees. This is equal to 41 degrees. And this is also true for angle 𝐴𝐶𝐵.
At this stage, it may not be obvious where we go next. However, if we let the point of intersection of line segment 𝐵𝐶 and line segment 𝑀𝐴 be 𝑋, we can consider the four triangles shown. After clearing some space, we will consider why some of these triangles are congruent. It is clear from the diagram that triangles one and two, 𝑀𝐵𝑋 and 𝑀𝐶𝑋, are congruent. This is because they satisfy the side-side-angle property. They have two sides that are equal in length together with an angle. The same is true for triangles three and four. Triangles 𝐴𝐵𝑋 and 𝐴𝐶𝑋 are congruent due to the side-side-angle property. The side length 𝐴𝐵 is equal to the side length 𝐴𝐶. Both triangles contain the side 𝐴𝑋, and angle 𝐴𝐵𝑋 is equal to angle 𝐴𝐶𝑋.
The angles 𝐵𝑋𝐴 and 𝐶𝑋𝐴 must sum to 180 degrees, as they lie on a straight line. Also, since the triangles are congruent, these angles must be equal. This means that they’re equal to 90 degrees. Triangles 𝐴𝐵𝑋 and 𝐴𝐶𝑋 are congruent right triangles.
We are now in a position to calculate the measure of angle 𝐵𝐴𝑋. This will have the same measure as angle 𝐵𝐴𝑀. Since angles in a triangle sum to 180 degrees, we have the measure of angle 𝐵𝐴𝑋 plus 90 degrees plus 41 degrees is equal to 180 degrees. Subtracting 90 degrees and 41 degrees from both sides of our equation, we have the measure of angle 𝐵𝐴𝑋 is equal to 49 degrees. We can therefore conclude that the measure of angle 𝐵𝐴𝑀 is 49 degrees.
Whilst it is not required in this question, this means that angle 𝐶𝐴𝑋 is also 49 degrees. And using the same method with the congruent triangles 𝑀𝐵𝑋 and 𝑀𝐶𝑋, we see that angle 𝐵𝑀𝑋 and 𝐶𝑀𝑋 are both 41 degrees. This means that we have four similar triangles with three equal angles and shape 𝑀𝐵𝐴𝐶 is a kite.