Video Transcript
Circle 𝑀 has radius 65. Suppose 𝐴 is on a line 𝐿 and line segment 𝑀𝐴 is perpendicular to 𝐿. If two 𝑀𝐴 minus 56 equals 18, what can be said of how 𝐿 lies with respect to the circle? (A) 𝐿 is a tangent to circle 𝑀, (B) 𝐿 is secant to circle 𝑀, or (C) 𝐿 is outside of circle 𝑀.
Let’s use the information given in the question to sketch the scenario. Here is circle 𝑀, and we’re told that its radius is 65 length units. This means suppose we have a point 𝑃 at any location on the circumference of the circle, line segment 𝑀𝑃, that’s the radius of the circle, is 65 length units. We then have this line 𝐿. Point 𝐴 lies on this line so that the line segment between 𝑀 and 𝐴 is perpendicular to the line. So, there are three possible options for our line. Either the whole line lies outside of the circle as shown. Point 𝐴 could in fact lie on the circumference. Alternatively, the line could pass through the circle, meaning that point 𝐴 lies somewhere on the radius.
In order to figure out which of these scenarios it is, we can work out the length of line segment 𝑀𝐴. If this dimension is shorter than the radius, if it’s less than 65 length units, then 𝐴 must lie inside the circle as we see. If it’s equal to the radius, if it’s exactly 65 length units, then point 𝐴 lies on the circumference and 𝐿 is a tangent. And finally, if 𝑀𝐴 is longer than the radius, if it’s greater than 65, then we know 𝐴 and the line 𝐿 lie outside of the circle. So, let’s solve the equation two 𝑀𝐴 minus 56 equals 18.
We’re going to add 56 to both sides. On the left-hand side, that leaves us with simply two 𝑀𝐴. On the right-hand side, 18 plus 56 equals 74. So two 𝑀𝐴 equals 74. And we can solve for 𝑀𝐴 by dividing through by two. So, 𝑀𝐴 is 74 divided by two, which is 37. So, 𝑀𝐴 is 37 length units. If 𝑀𝐴 is 37, this means it’s less than 65 length units. 𝑀𝐴 is less than the length of the radius. This means we choose this left-most line for 𝐿. 𝐿 must therefore pass through the circle at exactly two points. A line that passes through the circle exactly twice, in fact, is called a secant.
So, the correct answer is (B). 𝐿 is secant to circle 𝑀.
