For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the midpoint M of line segment PQ. See Examples 1 and 2. P(-5, -6), Q(7, -1)

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Identify the coordinates of points P and Q: P has coordinates (-5, -6) and Q has coordinates (7, -1).

To find the distance d(P, Q) between points P and Q, use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] where \((x_1, y_1)\) are the coordinates of P and \((x_2, y_2)\) are the coordinates of Q.

Substitute the coordinates into the distance formula: \[d = \sqrt{(7 - (-5))^2 + (-1 - (-6))^2}\] and simplify inside the parentheses.

To find the midpoint M of the line segment PQ, use the midpoint formula: \[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of P and Q respectively.

Substitute the coordinates into the midpoint formula: \[M = \left( \frac{-5 + 7}{2}, \frac{-6 + (-1)}{2} \right)\] and simplify the expressions inside the parentheses to find the midpoint coordinates.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distance Formula

The distance formula calculates the length between two points in the coordinate plane. It is derived from the Pythagorean theorem and given by d = √[(x2 - x1)² + (y2 - y1)²], where (x1, y1) and (x2, y2) are the coordinates of the points.

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points. It is calculated by averaging the x-coordinates and y-coordinates separately: M = ((x1 + x2)/2, (y1 + y2)/2). This gives the coordinates of the midpoint on the line segment.

Coordinate Geometry Basics

Coordinate geometry connects algebra and geometry using the coordinate plane. Understanding how points, lines, and distances relate through coordinates is essential for solving problems involving distances and midpoints between points.

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