Prove that tangent is perpendicular to radius all answer states that angle opp to largezt side is greatest but not 90
Answers
Answer:
Step-by-step explanation:
There’s a fairly simple proof for this. To do this, you first need to prove that the point of intersection between the tangent line and the circle is the point on the tangent that is closest to the center of the circle. To get to any other point on the tangent line from the center of the circle, you need to go the length of the radius…and then some more, depending on where on the tangent you are trying to get to. (This is easy to see if you draw a picture or visualize it). Now, forget about the circle and the tangent line. Just consider some random line in all of space. Now think of a point off of that line. Now, what you need to do is prove that that point off of the line is perpendicular to the line WHEN it is connected with a line segment to the point ON the line that it is closest to. Then, after we can prove this, since we already said that the point of intersection between a tangent and the circle is the point of the tangent line closest to the center of the circle, we will be done. There are a lot of ways to prove the perpendicular to closest point on line idea, but I think the best way is by contradiction. So if the point is NOT perpendicular to the line when connected to the closest point on the line, that means that the two angles formed by the point and the line are not equal to 90 degrees. However, this contradicts what we said earlier: if that point on the line that we connected the other point to is the closest to that point, then there should be a straight line connecting the two. This is because the shortest distance between two points is always a straight line. This is the simple, logical way to prove it, but you can also construct a right triangle on the line and with the point off the line to find a shorter distance between the point and the line. The original line connecting the two points would be the hypotenuse, and one of the legs would be connecting the point off the line to some point other than the point on the line that we already made. By definition and the Pythagorean Theorem, this leg is shorter than the hypotenuse, so the shortest distance between the point and the line will be achieved when the line segment connecting the two is perpendicular to the line.