Video Transcript
Given that the measure of angle ππΆπ΅ is equal to 49 degrees, where line segments π΄π΅ and π΄πΆ are tangent to the circle at π΅ and πΆ, find the measure of angle π΅π΄π.
We are told in the question that the measure of angle ππΆπ΅ is 49 degrees. And we are asked to find the measure of angle π΅π΄π. We are also told that the line segments π΄π΅ and π΄πΆ 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 π΄π΅ and π΄πΆ 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 ππ΅π΄ and ππΆπ΄ are 90 degrees. Since ππ΅ and ππΆ are radii to the circle, they must be equal in length. This means that triangle ππ΅πΆ is isosceles. And since two angles in an isosceles triangle are equal, the measure of angle ππ΅πΆ is 49 degrees. The measure of angle π΄π΅πΆ is therefore equal to 90 degrees minus 49 degrees. This is equal to 41 degrees. And this is also true for angle π΄πΆπ΅.
At this stage, it may not be obvious where we go next. However, if we let the point of intersection of line segment π΅πΆ and line segment ππ΄ be π, 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, ππ΅π and ππΆπ, 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 π΄π΅π and π΄πΆπ are congruent due to the side-side-angle property. The side length π΄π΅ is equal to the side length π΄πΆ. Both triangles contain the side π΄π, and angle π΄π΅π is equal to angle π΄πΆπ.
The angles π΅ππ΄ and πΆππ΄ 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 theyβre equal to 90 degrees. Triangles π΄π΅π and π΄πΆπ are congruent right triangles.
We are now in a position to calculate the measure of angle π΅π΄π. This will have the same measure as angle π΅π΄π. Since angles in a triangle sum to 180 degrees, we have the measure of angle π΅π΄π 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 π΅π΄π is equal to 49 degrees. We can therefore conclude that the measure of angle π΅π΄π is 49 degrees.
Whilst it is not required in this question, this means that angle πΆπ΄π is also 49 degrees. And using the same method with the congruent triangles ππ΅π and ππΆπ, we see that angle π΅ππ and πΆππ are both 41 degrees. This means that we have four similar triangles with three equal angles and shape ππ΅π΄πΆ is a kite.
