Question Video: Finding the Length of a Right-Angled Triangle’s Side given the Others Two Sides’ Lengths Using the Tangents’ Properties Mathematics

Line 𝐴𝐢 is tangent to circle 𝑀 at 𝐴. Given that 𝐡𝑀 = 55 cm, 𝐴𝐢 = 96 cm, what is 𝐡𝐢?

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Video Transcript

Line 𝐴𝐢 is tangent to circle 𝑀 at 𝐴. Given that 𝐡𝑀 equals 55 centimeters, 𝐴𝐢 equals 96 centimeters, what is 𝐡𝐢?

Let’s begin by adding what we know about our circle and the lines within it to the diagram itself. First, we’re given the length of the line segment between 𝐡 and 𝑀. It’s 55 centimeters. We’re also told that the length of 𝐴𝐢 is 96 centimeters, and we’re asked to calculate 𝐡𝐢. That’s the length of the line segment that joins 𝐡 to 𝐢. So let’s define that to be equal to π‘₯ centimeters. Now, we also know that line 𝐴𝐢 is a tangent to the circle at point 𝐴. And we also see that line segment 𝐡𝐴 passes through point 𝑀. This means that line segment 𝐡𝐴 must be the diameter of the circle.

So what do we know about the relationship between a diameter and tangent of a circle? In fact, they are perpendicular to one another. In other words, angle 𝐡𝐴𝐢 is equal to 90 degrees. So triangle 𝐴𝐡𝐢 is in fact a right triangle. And of course, we know that if we know two of the dimensions of a right triangle, we can calculate the third by using the Pythagorean theorem. And whilst it might not look like it, we do actually know two of the dimensions in this triangle. Remember, we were told 𝐡𝑀 is equal to 55 centimeters. 𝐡𝑀 is in fact the radius of the circle. This means line segment 𝐡𝐴 must be double the length of line segment 𝐡𝑀 since 𝐡𝐴 is the diameter. Well, twice 55 centimeters is 110 centimeters, so line segment 𝐡𝐴 is 110 centimeters in length.

We can therefore use the Pythagorean theorem to find the length that we’ve called π‘₯ centimeters. The Pythagorean theorem says that, for a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In this case, the hypotenuse is equal to π‘₯ centimeters. So 110 squared plus 96 squared is equal to π‘₯ squared. 110 squared is 12100, whilst 96 squared is 9216. Finding their sum and we see that π‘₯ squared is equal to 21316.

To solve this equation for π‘₯, we find the square root of both sides. And remember, since π‘₯ represents a dimension, we’re only interested in the positive square root of 21316. The positive square root of this number is in fact 146. So we found that π‘₯ is equal to 146. We can therefore say that 𝐡𝐢 equals 146 centimeters.

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