Problem 92 - A Graphical Approach to College Algebra

Question

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(-6,-10), Q(6,5)$$

Step-by-Step Solution

Answer

Distance: \(\sqrt{369}\); Midpoint: \((0, -2.5)\).

Step 1: Identify Coordinates

Identify the coordinates of points \(P\) and \(Q\). Point \(P\) has coordinates \((-6,-10)\) and point \(Q\) has coordinates \((6,5)\).

Step 2: Apply Distance Formula

The distance formula is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Use the coordinates of \(P(x_1, y_1) = (-6, -10)\) and \(Q(x_2, y_2) = (6, 5)\) to find the distance.

Step 3: Calculate Differences

Calculate the differences in the x-coordinates and y-coordinates. The difference in x-coordinates is \(x_2 - x_1 = 6 - (-6) = 12\). The difference in y-coordinates is \(y_2 - y_1 = 5 - (-10) = 15\).

Step 4: Compute Distance

Plug the differences into the distance formula: \[d = \sqrt{12^2 + 15^2} = \sqrt{144 + 225} = \sqrt{369}\]. Therefore, the distance \(d\) is \(\sqrt{369}\).

Step 5: Apply Midpoint Formula

The midpoint formula is \((x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Use the coordinates of \(P\) and \(Q\) to find the midpoint.

Step 6: Calculate Midpoint Coordinates

Compute the midpoint by calculating each component: \(x_m = \frac{-6 + 6}{2} = 0\) and \(y_m = \frac{-10 + 5}{2} = -2.5\). Hence, the coordinates of midpoint \(M\) are \((0, -2.5)\).

Key Concepts

Midpoint Formula

To find the point that lies exactly halfway between two given points in a coordinate plane, you use the midpoint formula. This formula gives you the average position between these two points. It's like finding the center spot when you fold a string into two equal halves. The midpoint formula is:\[ (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]It involves adding up the x-coordinates of the two points, dividing by two, and doing the same for the y-coordinates.

• For example, if you have points \( P(-6, -10) \) and \( Q(6, 5) \), the average of their x-coordinates is \( \frac{-6 + 6}{2} = 0 \) and for the y-coordinates, it is \( \frac{-10 + 5}{2} = -2.5 \).
• Thus, the midpoint M is located at \( (0, -2.5) \).

By using the midpoint formula, you can easily find a point equally distant from two others, which is especially useful in geometry when bisecting line segments. You can visualize it like setting up a seesaw balanced perfectly at the midpoint.

Coordinate Geometry

Coordinate geometry, also known as Cartesian geometry, is the study of geometric figures through the use of a coordinate plane. This mathematical domain provides tools for understanding the spatial relationship between points using their coordinates \( (x, y) \).
In this system, each point on the plane is specified by an orderly pair of numbers. When dealing with problems in coordinate geometry:

• It becomes immensely practical because you can apply formulas like the distance or midpoint formulas to find or verify various properties of shapes and figures.
• For instance, with points \( P(-6, -10) \) and \( Q(6, 5) \), tasks such as finding distances, midpoints, or even slopes rely on the coordinates provided.

Coordinate geometry acts as a bridge between algebra and geometry, allowing numerical solutions and graphical understanding of problems. It is fundamental in many areas of mathematics and engineering because it reveals how equations correspond to geometric shapes.

Distance Calculation

Calculating distance between two points in a coordinate plane is like measuring the space between them, accounting for both their horizontal and vertical differences. The distance formula is derived from the Pythagorean theorem and expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
• \( x_2 - x_1 \) is the horizontal distance, and \( y_2 - y_1 \) is the vertical distance between these points.

For example, given points \( P(-6, -10) \) and \( Q(6, 5) \):

• The horizontal difference \( 6 - (-6) \) is \( 12 \).
• The vertical difference \( 5 - (-10) \) is \( 15 \).
• Substitute these into the formula to get \( \sqrt{12^2 + 15^2} = \sqrt{369} \).

This gives you the straight-line distance between these two points. Understanding and being able to apply the distance formula is key to finding precise measurements in coordinate geometry. It enables you to view distances visually and numerically across dimensions.