Question Video: Finding the Length of a Side in a Right Triangle Using the Pythagorean Theorem Mathematics

Video Transcript

A circle of center 𝑀 has a radius of 11 centimeters. If 𝐶𝐴 is 16.3 centimeters, what is line segment 𝐴𝐵? Give your answer to the nearest tenth.

We’re given a diagram of a circle and then a triangle formed by connecting three points: 𝑀, the center of the circle; 𝐵, which is a point on the circumference; and 𝐴, which is an external point. We want to calculate the length of line segment 𝐴𝐵. We know that 𝐶𝐴 equals 16.3 centimeters. We’re also told that the radius is 11 centimeters. Remember, the radius of a circle is the line whose endpoints are the center of the circle and a point on the circumference. This means both 𝑀𝐵 and 𝑀𝐶 are radii of this circle. And both measure 11 centimeters.

We now know the lengths of two out of the three sides in this triangle. We also notice here that the line 𝐴𝐵 is tangent to the circle at point 𝐵. This means line 𝐴𝐵 is tangent to the circle at one of its radii. If we remember that a tangent to a circle is perpendicular to the radius at the point of contact, this tells us that angle 𝐴𝐵𝑀 is a right angle. And that makes triangle 𝐴𝐵𝑀 a right triangle. And if we know the lengths of two sides of a right triangle and we want to calculate the length of a third side, we can do so by applying the Pythagorean theorem.

The Pythagorean theorem says that 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎 and 𝑏 are the smaller side lengths and 𝑐 is the length of the hypotenuse. In our given triangle, the hypotenuse, the side opposite the right angle, will be the line segment 𝑀𝐴, which will be 11 plus 16.3, 27.3 centimeters. We can let 𝑎 squared be equal to the radius that we know, 𝑀𝐵 squared. And in place of the 𝑏 squared value, we’ll substitute our missing side, 𝐴𝐵, squared. We plug in the values we know for 𝑀𝐵 and 𝑀𝐴. Then we’ll square those values. By subtracting 121 from both sides, we find that 𝐴𝐵 squared equals 624.29. And finally, we’ll take the square root of both sides. Line segment 𝐴𝐵 will be equal to the square root of 624.29.

We want this value to the nearest tenth. Plugging this into a calculator gives us the square root equal to 24.98579 continuing. To round to the nearest tenth, we look to the right of the tenths place. This means we need to round up. Since there’s a nine in the tenths place, to round up, we need to go up one whole unit, making this 25.0. And this is a measure of centimeters. The missing line segment 𝐴𝐵 is equal to 25.0 centimeters.