Question

diagnostic assessment tangents intersecting a radius line l represents a straight part of the shoreline at a beach. suppose you are in the ocean at point c and you want to get to the shore as fast as possible. assume there is no current. segments cj and cd represent 2 possible paths. diego says,
o matter where we put point d, the pythagorean theorem tells us that segment cj is shorter than segment cd. so, segment cj represents the shortest path to shore.\ do you agree with diego? explain your reasoning.

Explanation:

The shortest distance from a point to a line is the perpendicular distance. If \(CJ\) is perpendicular to line \(\ell\) (the shore - line), then by the Pythagorean theorem, for any other point \(D\) on \(\ell\), in right - triangle \(CJD\) (where \(\angle CJD = 90^{\circ}\)), \(CD^{2}=CJ^{2}+JD^{2}\). Since \(JD^{2}>0\) for \(D
eq J\), \(CD > CJ\). So, if \(CJ\) is the perpendicular from point \(C\) to line \(\ell\), Diego is correct.

Answer:

Yes, if \(CJ\) is perpendicular to the shore - line \(\ell\). The Pythagorean theorem states that in a right - triangle with hypotenuse \(CD\) and legs \(CJ\) and \(JD\) (where \(CJ\) is the perpendicular distance from \(C\) to \(\ell\)), \(CD^{2}=CJ^{2}+JD^{2}\), so \(CD>CJ\) for \(D
eq J\).