The maximum value of $PC$ occurs when the point $C$ is located on the opposite side of the triangle from the point $A$. In this case, $PC$ is equal to the length of the longest side of the triangle, also known as the hypotenuse.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In $\triangle ABC$, $PC$ is the hypotenuse, so we can use the Pythagorean theorem to find its maximum value by squaring the lengths of sides $AB$ and $AC$, then taking the square root of the sum.
No, the maximum value of $PC$ cannot be greater than the sum of the lengths of sides $AB$ and $AC$. This is because in a triangle, any side must be shorter than the sum of the other two sides. Therefore, the hypotenuse $PC$ cannot be longer than the sum of $AB$ and $AC$.
The maximum value of $PC$ is directly related to the angles of $\triangle ABC$. In a right triangle, the length of the hypotenuse is always greater than the lengths of the other two sides, and this relationship holds true for any type of triangle. As the angles of $\triangle ABC$ increase, the maximum value of $PC$ also increases, and vice versa.
The position of point $C$ greatly affects the maximum value of $PC$ in $\triangle ABC$. As mentioned earlier, when $C$ is located on the opposite side of the triangle from $A$, $PC$ is at its maximum value. However, if $C$ is located on the same side as $A$, then $PC$ will be shorter. In general, the closer $C$ is to $A$, the shorter $PC$ will be, and the farther away $C$ is, the longer $PC$ will be.
