Video Transcript
Given that the line segment 𝐴𝐵 is tangent to the circle with center 𝑀 at 𝐴, 𝐴𝑀
equals 8.6 centimeters, and 𝑀𝐵 equals 12.3 centimeters, find the length of the
line segment 𝐴𝐵, and round the result to the nearest tenth.
Let’s begin by adding the information given in the question onto the diagram. 𝐴𝑀 is 8.6 centimeters long. 𝑀𝐵 is 12.3 centimeters. And the length we’re looking to find is the length of the line segment 𝐴𝐵. Now we notice that we have a triangle, triangle 𝐴𝑀𝐵, for which we know the lengths
of two of the sides. Our first thought then may be that we can apply the Pythagorean theorem. But remember the Pythagorean theorem is only valid in right triangles. So we need to consider whether the triangle 𝐴𝑀𝐵 is a right triangle or not.
The other key piece of information given in the question is that the line segment
𝐴𝐵 is tangent to the circle at the point 𝐴. A key property about tangents to circles is that a tangent to a circle is
perpendicular to the radius of the circle at the point of contact. So the line segment 𝐴𝐵 is perpendicular to the radius 𝐴𝑀. And we therefore have a right angle at 𝐴 in our triangle 𝐴𝑀𝐵. And so the measure of angle 𝑀𝐴𝐵 is 90 degrees, and we do indeed have a right
triangle. So we can apply the Pythagorean theorem to calculate the length of the third
side.
The Pythagorean theorem tells us that in a right triangle, the sum of the squares of
the two shorter sides is equal to the square of the longest side of the
triangle. Remember, the longest side, or hypotenuse, is always the side directly opposite the
right angle. So, in this case, that’s the side 𝑀𝐵, which is 12.3 centimeters long. Substituting 𝐴𝐵 and 8.6 for the two shorter sides of the triangle and 12.3 for the
longest side, or hypotenuse, we have the equation 𝐴𝐵 squared plus 8.6 squared is
equal to 12.3 squared. We can evaluate 8.6 squared and 12.3 squared and then subtract 73.96, that’s 8.6
squared, from both sides, giving 𝐴𝐵 squared equals 77.33.
We solve this equation by taking the square root on both sides. And we’re only going to take the positive value here as 𝐴𝐵 is the length of a side
of a triangle, which is a positive quantity. Evaluating this square root on a calculator, we find that 𝐴𝐵 is equal to 8.79374
and so on. Remember though that we’re asked to round the result to the nearest tenth, which is
to one decimal place. And as there’s a nine in the hundredths column, we round up, giving 𝐴𝐵 equal to 8.8
centimeters.
