Video: Finding the Measure of an Inscribed Angle by Using the Properties of Tangents to the Circle

Given that the line segment 𝐴𝐡 is a tangent to the circle 𝑀, and π‘šβˆ π΄π΅π‘€ = 49Β°, determine π‘šβˆ π΄π·π΅.

03:49

Video Transcript

Given that the line segment 𝐴𝐡 is a tangent to the circle 𝑀, and the measure of angle 𝐴𝐡𝑀 is 49 degrees, determine the measure of angle 𝐴𝐷𝐡.

Angle 𝐴𝐷𝐡 is the angle formed when we travel from 𝐴 to 𝐷 to 𝐡. So, it’s the angle marked in orange on the diagram. Angle 𝐴𝐡𝑀 is the angle formed when we travel from 𝐴 to 𝐡 to 𝑀. It’s the angle now marked in pink on the diagram with its measure of 49 degrees. From the information given, we aren’t able to calculate angle 𝐴𝐷𝐡 directly. We’re going to need to find the measures of some other angles in the figure first. The other key piece of information given in the question, though, is that the line 𝐴𝐡 is a tangent to the circle 𝑀. And the key property about tangents of circles is that a tangent to a circle is perpendicular to the radius at the point of contact.

The point where the tangent meets the circle is point 𝐴. And the radius here is the line segment 𝐴𝑀. So, we know that the angle 𝐡𝐴𝑀 is 90 degrees. So, we now know one more angle within the figure. We still aren’t able to calculate angle 𝐴𝐷𝐡 directly. So, let’s see what other angles we could work out. We have a triangle. In fact, it’s a right triangle, triangle 𝐴𝑀𝐡. And we know two of its angles, the right angle and the angle of 49 degrees. So, using the fact that angles in a triangle sum to 180 degrees, we can calculate the third angle in this triangle.

We have that angle 𝐴𝑀𝐡 plus 90 degrees plus 49 degrees equals 180 degrees. 90 plus 49 is 139. And subtracting this from 180, we find that angle 𝐴𝑀𝐡 is 41 degrees. So, we now know another angle in our diagram. We still don’t have enough information to calculate angle 𝐴𝐷𝐡. But we can now calculate a different angle, angle 𝐴𝑀𝐷. We know that the angles on any straight line sum to 180 degrees. So, angle 𝐴𝑀𝐷 plus the angle we’ve just calculated of 41 degrees must equal 180 degrees. Angle 𝐴𝑀𝐷 is, therefore, equal to 180 degrees minus 41 degrees. That’s 139 degrees.

Now, we found almost all of the angles in the figure, but still not the one that we were looking for. The final step is to consider triangle 𝐴𝑀𝐷, in which we know one angle of 139 degrees. We need to notice that both 𝑀𝐷 and 𝑀𝐴 are radii of the circle 𝑀. And therefore, they’re of the same length. This means that triangle 𝑀𝐷𝐴 is an isosceles triangle. And it also means that angle 𝑀𝐷𝐴 will be equal to angle 𝑀𝐴𝐷. We can, therefore, find the measure of each angle by subtracting the third angle, 139 degrees, from the total angle sum in a triangle, 180 degrees, and then halving the remainder. Doing so gives each of these angles to be 20.5 degrees. Now, angle 𝑀𝐷𝐴 is in fact the same angle as angle 𝐴𝐷𝐡. They both refer to this angle here. And so, we’ve completed the problem.

By using some of the more basic facts about angles in triangles and angles in straight lines and then the key property that a tangent to a circle is perpendicular to the radius at the point of contact, we’ve found the measure of angle 𝐴𝐷𝐡 is 20.5 degrees.

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