Question Video: Using the Properties of Tangency to Find Chord Lengths and Calculate the Are of a Triangle inside a Circle Mathematics • 11th Grade

Video Transcript

Triangle 𝐴𝐵𝐶 is right angled at 𝐵. Given that the measure of angle 𝐴 is 30 degrees, that 𝐴𝐵 is equal to 37.6 centimeters, and 𝑀 is the circle that lies on 𝐴 and 𝐵 and is tangent to line segment 𝐴𝐶 at 𝐴, determine the area of triangle 𝐴𝐵𝑀 to the nearest tenth of a square centimeter.

We’ve been given this whole description without an image. Our first job here is going to be to sketch an image to represent the information we’ve been given. So we start at the beginning. We have triangle 𝐴𝐵𝐶, which has a right angle at 𝐵. And we know that the measure of angle 𝐴 is 30 degrees. Line segment 𝐴𝐵 measures 37.6 centimeters. The next thing we know is that there is some circle 𝑀 that lies on 𝐴 and 𝐵. That means that 𝐴𝐵 is a chord of the circle.

But we especially know that the line segment 𝐴𝐶 is a tangent to the circle, which means we can sketch something like this. 𝑀 is the center of that circle. And then we know that the points 𝐴, 𝐵, and 𝑀 make a triangle. From here, we’ll need to use our knowledge of both circles and triangles to figure out this area. The area of triangle 𝐴𝐵𝑀 is what we want to find. We know to find the area of a triangle, we have one-half the height times the base. The base of this triangle is 37.6. It’s the line segment 𝐴𝐵. So we can go ahead and add that value. The height of this triangle is the perpendicular distance from point 𝑀 to the base. And that’s the missing value we need to solve for.

But before we can do that, we’ll need to fill in more information. Now, we know that line segment 𝐴𝐶 is tangent to this circle at point 𝐴. And that means it forms a right angle. We already know that the angle 𝐵𝐴𝐶 is 30 degrees. And that will make this remaining portion of the right angle 60 degrees because together they form a right angle, which is 90 degrees. Now, if 𝑀𝐴𝐵 measures 60 degrees, we know that 𝐴𝐵𝑀 will measure 60 degrees as well. We know this because they’re both opposite radii of the circle, which means line segment 𝐴𝑀 is equal in length to line segment 𝐵𝑀. It also means the triangle we’re trying to find the area of is an equilateral triangle, 60 degrees on all three angles.

But again, in order for us to find the height of this triangle, we need to know this distance. To find that, let’s try and zoom in on this portion of our figure. We have a right angle and a 60-degree angle, which means the remaining angle will be 30 degrees. If we call this point 𝑑, when we go back to our original figure, we can say that 𝐴𝑑 is going to be equal in length to 𝑑𝐵. This is because the line segment 𝑀𝑑, the height of this triangle, is a perpendicular bisector. And that means the segment 𝑑𝐵 is half 37.6, which is 18.8 centimeters. Looking at our triangle here, we know all the angles and we know one side length.

In addition to that, we can say that this is a special right triangle, a 30-60-90 triangle. And we know that the ratio of side lengths in a 30-60-90 triangle occur in the ratio one to square root of three to two. In this case, the side length opposite 30 degrees measures 18.8. And the side length opposite the 60-degree angle is the value we’re interested in, the height of this triangle. To go from one to the square root of three, you multiply by the square root of three, which means the height of our triangle will be equal to 18.8 times the square root of three, which becomes 32.5625 continuing. And that’s a measure of centimeters.

If we round to three decimal places, we get a height of 32.563 centimeters. So we take that value, and we plug it in to our area formula. Then we enter this into our calculator. And we get 612.1844. And this is area, so it is a measure of centimeters squared. We’re looking for this rounded to the nearest tenth of a square centimeter. We look to the right of that, and we have an eight, which means we will round up to 612.2 centimeters squared. The area of triangle 𝐴𝐵𝑀 is 612.2 centimeters squared.