Question Video: Relating Points- Lines- and Circles in Space Mathematics

Consider a circle of center 𝑀 with a radius π‘Ÿ and a point 𝐴 on a line 𝐿, where the line 𝐿 does not intersect the circle and line segment 𝑀𝐴 βŠ₯ 𝐿. Which of the following is true? [A] 𝑀𝐴 = π‘Ÿ [B] 𝑀𝐴 < π‘Ÿ [C] 𝑀𝐴 > π‘Ÿ

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Video Transcript

Consider a circle of center 𝑀 with a radius π‘Ÿ and a point 𝐴 on a line 𝐿, where the line 𝐿 does not intersect the circle and line segment 𝑀𝐴 is perpendicular to 𝐿. Which of the following is true? Is it (A) 𝑀𝐴 equals π‘Ÿ, (B) 𝑀𝐴 is less than π‘Ÿ, or (C) 𝑀𝐴 is greater than π‘Ÿ?

To answer this question, let’s begin by drawing a sketch of the scenario. Here’s our circle center 𝑀. We’re given that it has a radius π‘Ÿ units. In other words, consider a point 𝑃 that lies anywhere on the circumference of the circle. The line segment joining the center of the circle to this point is π‘Ÿ units. Then, we have this line 𝐿 that doesn’t intersect the circle. Point 𝐴 lies on the line, and we’re told that the line segment 𝑀𝐴 is perpendicular to 𝐿. And so now we should be able to answer the question quite quickly.

We said that the length of line segment 𝑀𝑃 is π‘Ÿ units. Let’s define the length of the line segment between 𝑀 and 𝐴 to be π‘₯ units. And we can quite quickly see just from the diagram alone that π‘₯ must be greater than π‘Ÿ. Since π‘₯ is the length of line segment 𝑀𝐴, then we’re able to say that 𝑀𝐴 is greater than π‘Ÿ. And so the answer is (C): 𝑀𝐴 is greater than π‘Ÿ.

Now, in fact, this will be true for any point 𝐴 that lies outside of the circle. Since the radius is the length of the line segment that joins the center to a point on its circumference, if 𝐴 lies outside of the circle in any position, then it makes sense that the line segment between the center and 𝐴 is greater than the radius.

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