Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (-9, 8), endpoint (-16, 9)

검증된 단계별 안내

Recall the midpoint formula for a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ \left( \frac{ x_1 + x_2 }{2}, \frac{ y_1 + y_2 }{2} \right) = (x_m, y_m) \] where \( (x_m, y_m) \) is the midpoint.

Identify the known values from the problem: the midpoint \( (x_m, y_m) = (-9, 8) \) and one endpoint \( (x_1, y_1) = (-16, 9) \). The other endpoint \( (x_2, y_2) \) is unknown.

Set up two equations using the midpoint formula by equating the midpoint coordinates to the average of the endpoints' coordinates: \[ \frac{ -16 + x_2 }{2} = -9 \] \[ \frac{ 9 + y_2 }{2} = 8 \]

Solve each equation for the unknown coordinate: For the \(x\)-coordinate: \[ -16 + x_2 = 2 \times (-9) \] \[ x_2 = 2 \times (-9) + 16 \] For the \(y\)-coordinate: \[ 9 + y_2 = 2 \times 8 \] \[ y_2 = 2 \times 8 - 9 \]

Write the coordinates of the other endpoint as \( (x_2, y_2) \) using the expressions found in the previous step.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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: Midpoint = ((x1 + x2)/2, (y1 + y2)/2). This formula is essential for relating endpoints to their midpoint.

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: x2 = 2 * midpoint_x - x1, y2 = 2 * midpoint_y - y1. This method helps find the missing endpoint.

Coordinate Geometry

Coordinate geometry involves using algebraic methods to solve geometric problems on the coordinate plane. Understanding how points, lines, and segments relate through their coordinates allows for precise calculations like finding midpoints and endpoints, which is fundamental in this problem.