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Line represents a straight part of the shoreline at a beach. Suppose you are in the ocean at point and you want to get to the shore as fast as possible. Assume there is no current. Segments and represent two possible paths.
Diego says, “No matter where we put point , the Pythagorean Theorem tells us that segment is shorter than segment . So segment represents the shortest path to shore.”
Do you agree with Diego? Explain your reasoning.
The image shows an angle whose rays are tangent to a circle.
A line is said to be tangent to a circle if it intersects the circle at exactly 1 point. Suppose line is tangent to a circle centered at . Draw a radius from the center of the circle to the point of tangency, or the point where line intersects the circle. Call this point . It looks like radius is perpendicular to line . Can we prove it?
Every other point on the tangent line is outside the circle, so they must all be further away from the center than the point where the tangent line intersects the circle. This means the point where the tangent line intersects the circle is the closest point on the line to the center point . The radius must be perpendicular to the tangent line because the shortest distance from a point to a line is always along a perpendicular path.
A line is tangent to a circle if the line intersects the circle at exactly one point and is perpendicular to the radius.