Video: Finding the Length of a Tangent to a Circle Using the Application of Similarity in Circles

In the figure, the circle has a radius of 12 cm, 𝐴𝐡 = 12 cm, and 𝐴𝐢 = 35 cm. Determine the distance from line segment 𝐡𝐢 to the center of the circle, 𝑀, and the length of line segment 𝐴𝐷, rounding your answers to the nearest tenth.

03:51

Video Transcript

In the figure shown, the circle has a radius of 12 centimeters, 𝐴𝐡 equals 12 centimeters, and 𝐴𝐢 equals 35 centimeters. Determine the distance from line segment 𝐡𝐢 to the center of the circle 𝑀 and the length of line segment 𝐴𝐷, rounding your answers to the nearest tenth.

The first thing we wanna do is take information in our question and use it to label our diagram. We know the circle has a radius of 12 centimeters. That will be the distance from 𝑀 to any point on the outside of the circle. We also know that 𝐴𝐡 is 12 centimeters and 𝐴𝐢 equals 35 centimeters. Now, this also means we can find out what the distance from 𝐡 to 𝐢 will be. And the distance from 𝐡 to 𝐢 must be 35 minus 12, so it’s 23 centimeters. We want to find the distance from 𝐴 to 𝐷, and we want to find the distance from line segment 𝐡𝐢 to the middle of the circle. We’re looking for the shortest distance, and that’s going to be the perpendicular distance from the line segment 𝐡𝐢 to the center of the circle 𝑀.

Since line segment 𝐴𝐷 is a tangent and it intersects the secant 𝐴𝐢 at the point 𝐴, we can say that 𝐴𝐷 squared will be equal to 𝐴𝐡 times 𝐴𝐢. And we know that 𝐴𝐡 is 12 centimeters and 𝐴𝐢 is 35 centimeters, which means 𝐴𝐡 times 𝐴𝐢 is 420 centimeters squared. But to find 𝐴𝐷, we need to take the square root of both sides. We’re only interested in the positive square root, since we’re talking about distance. When we take the square root of 420, we get 20.4939 continuing. And we need to round that to the nearest tenth. When we do that, we find 𝐴𝐷 equals 20.5 centimeters.

Now, finding the distance from line segment 𝐡𝐢 to 𝑀 requires a little bit more information. We need to think about what we know about triangles. We know the distance from 𝑀 to 𝐢 because from 𝑀 to 𝐢 is a radius of this circle. We know that distance must be 12, which means the distance from 𝑀 to 𝐡 will also be 12 centimeters. What we’re now looking at is an isosceles triangle, where the distance from 𝑀 to the base is here. And the height of an isosceles triangle is also its median, which means it divides the base in half. So, we can divide this 23 centimeters by two, which will give us this distance to be 11.5 centimeters.

And then, we can use the Pythagorean theorem to find a missing side in a right-angled triangle. The Pythagorean theorem tells us that π‘Ž squared plus 𝑏 squared equals 𝑐 squared. In our case, the radius is the hypotenuse, so we need to square 12. We’ll let π‘Ž be our missing side length and 𝑏 be the 11.5 side length. So, we have π‘Ž squared plus 11.5 squared equals 12 squared. π‘Ž squared plus 132.25 equals 144. Then, we subtract 132.25 from both sides, and we get π‘Ž squared equals 11.75. Again, we need to take the square root. We only want the positive solution for distance. This gives us 3.4278 continuing, which we round to 3.4. The distance from line segment 𝐡𝐢 to the center of the circle is 3.4 centimeters, and 𝐴𝐷 equals 20.5 centimeters.

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