Question

1. \\( \overline { c d } \\) is the diameter of a circle whose center is the point \\( ( 2,1 ) \\). if the coordinates of \\( c \\) are \\( ( 0, - 2 ) \\), find the coordinates of \\( d \\). (endpoint) draw a sketch

Explanation:

Step1: Recall the mid - point formula

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let the coordinates of \(D\) be \((x,y)\). Since the center \((2,1)\) is the mid - point of \(C(0,-2)\) and \(D(x,y)\).

Step2: Set up equations for \(x\) and \(y\)

For the \(x\) - coordinate: \(\frac{0 + x}{2}=2\). Multiply both sides by \(2\): \(0 + x=4\), so \(x = 4\).
For the \(y\) - coordinate: \(\frac{-2 + y}{2}=1\). Multiply both sides by \(2\): \(-2 + y=2\). Add \(2\) to both sides: \(y=4\).

Answer:

The coordinates of \(D\) are \((4,4)\)