Question Video: The Properties of Tangents to a Circle Mathematics

A circle with center 𝑀 has a diameter line segment 𝐴𝐡. If line 𝐴𝐢 and line 𝐡𝐷 are two tangents to the circle, what can you say about them?

02:34

Video Transcript

A circle with center 𝑀 has a diameter line segment 𝐴𝐡. If line 𝐴𝐢 and line 𝐡𝐷 are two tangents to the circle, what can you say about them?

To solve this, let’s go ahead and sketch a circle. This is the circle 𝑀, with a diameter of 𝐴𝐡. We know that line segment 𝐴𝐡 must pass through the center 𝑀 since it is a diameter. And then we have a line 𝐴𝐢. We know that we name lines by two points that fall on that line. And because 𝐴𝐢 is tangent to this circle, it can only intersect the circle at point 𝐴. We can then sketch a line that looks like this for 𝐴𝐢. The line 𝐴𝐢 meets the diameter at a right angle. A similar thing is true for line segment 𝐡𝐷. We name the lines by points along that line. And because we know it’s tangent, it can only intersect the circle at point 𝐡. And so we have line 𝐡𝐷. This line also forms a right angle with the diameter.

What we’re seeing in this image is that line 𝐴𝐢 and line 𝐡𝐷 are parallel. The way we’ve drawn it, line 𝐴𝐢 and line 𝐡𝐷 are vertical and the diameter 𝐴𝐡 is horizontal. But this is not the only way we could draw the image. Let’s say we have the diameter drawn in this way. 𝐴𝐢 still forms a right angle with the diameter, as does 𝐡𝐷. And again, we’ll see that these two lines are parallel. They will never intersect. If both of these images still don’t convince you, we could do a short kind of proof.

If these lines are not parallel, then at some point in the distance, they will intersect. And we could call that point 𝑃. And if they intersect somewhere really far out in the distance, they would form a triangle. And the triangle would be 𝐴𝑃𝐡. The problem is we know that angles in a triangle must add up to 180 degrees. And since the measure of angle 𝐴 is 90 degrees and the measure of angle 𝐡 is 90 degrees, combined, they already equal 180 degrees. This confirms that these two lines can never intersect. They could not form a triangle and are therefore parallel. Line 𝐴𝐢 and line 𝐡𝐷 are parallel.

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