Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b).
midpoint $(6a,6b)$, endpoint $(3a,5b)$

Verified step by step guidance

1. Recall the midpoint formula for a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\): the midpoint \((x_m, y_m)\) is given by \(x_m = \frac{x_1 + x_2}{2}\) and \(y_m = \frac{y_1 + y_2}{2}\).
2. Identify the known values: the midpoint is \((6a, 6b)\) and one endpoint is \((3a, 5b)\). Let the other endpoint be \((x, y)\).
3. Set up equations using the midpoint formula: \(6a = \frac{3a + x}{2}\) and \(6b = \frac{5b + y}{2}\).
4. Multiply both sides of each equation by 2 to eliminate the denominators: \(2 \times 6a = 3a + x\) and \(2 \times 6b = 5b + y\).
5. Solve each equation for the unknown coordinate: \(x = 2 \times 6a - 3a\) and \(y = 2 \times 6b - 5b\). These expressions give the coordinates of the other endpoint.

Key Concepts

Midpoint Formula

The midpoint formula calculates the point exactly halfway between two endpoints of a line segment. It is found by averaging the x-coordinates and the y-coordinates of the endpoints separately, using ( (x1 + x2)/2, (y1 + y2)/2 ). This formula is essential for relating the midpoint to the endpoints.

Solving for an Unknown Endpoint

Given the midpoint and one endpoint, you can find the other endpoint by rearranging the midpoint formula. Multiply the midpoint coordinates by 2, then subtract the known endpoint coordinates to isolate the unknown endpoint. This method uses basic algebra to solve for missing values.

Coordinate Geometry with Variables

When coordinates involve variables (like a and b), treat them algebraically as you would numbers. Apply the midpoint formula and algebraic operations carefully, combining like terms and solving for unknown expressions. This approach allows solving problems with symbolic coordinates.