Problem 49 - Precalculus with Limits

Question

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \( (-4, 10) \), \( (4, -5) \)

Step-by-Step Solution

Answer

The distance between the points is 17 units and the midpoint of the line segment joining the points is (0, 2.5).

Step 1: Plotting the points

Plot the given points \( (-4, 10) \) and \( (4, -5) \) on the coordinate axis. The x-coordinate represents the horizontal placement and the y-coordinate represents the vertical placement of the points.

Step 2: Distance Calculation

Apply the distance formula which comes from the Pythagorean theorem. This is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the values we get:\( d = \sqrt{(4 - (-4))^2 + (-5 - 10)^2} = \sqrt{(8)^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17. Thus, the distance between the points is 17 units.

Step 3: Calculate the Midpoint

Apply the midpoint formula. The midpoint M of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substituting the values we get:\( M = \left( \frac{-4 + 4}{2}, \frac{10 + (-5)}{2} \right) = (0, \frac{5}{2}) = (0, 2.5). Thus, the midpoint of the line segment is (0, 2.5).

Key Concepts

Distance Formula

Understanding the distance formula is a pivotal part of coordinate geometry. It is a specific application of the Pythagorean theorem in a coordinate plane. The formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) calculates the straight-line distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \).

To apply this formula effectively, start by identifying the x and y coordinates of both points. Then, subtract the x-coordinates from each other, and the y-coordinates likewise, squaring both results. Finally, add these squares together and take the square root of the sum. This gives you the magnitude of the distance between the two points within the coordinate plane, much like measuring the hypotenuse of a right-angled triangle.

Midpoint Formula

The midpoint formula is equally important when dealing with a line segment in a coordinate plane. If you're tasked with finding the point that's exactly in the middle of a line segment connecting two points, this formula is your go-to tool. The coordinates of the midpoint M are found using the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

It requires you to simply average the x-coordinates of the two points for the x-coordinate of the midpoint, and do the same with the y-coordinates for the midpoint's y-coordinate. The resulting coordinates provide a precise value for the location of the midpoint, bridging the concepts of arithmetic mean and geometric space.

Coordinate Graphing

Coordinate graphing is a foundational concept in algebra and geometry that facilitates the visualization of numerical relationships. Using a two-dimensional grid with horizontal x-axis and vertical y-axis, points can be plotted according to their x (horizontal) and y (vertical) values. Each point is denoted by a pair of numbers \( (x, y) \) which indicate its position on the plane.

To graph points, begin at the origin (where the x and y axes intersect), and first move horizontally to the x-value, then vertically to the y-value. This visual representation makes it easier to understand concepts such as slope, distance, and midpoint, and allows for the practical application of algebra in geometrical form.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry that connects algebraic and geometric ideas. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be mathematically represented as \( a^2 + b^2 = c^2 \), where c represents the hypotenuse.

This theorem forms the basis for the distance formula, as the distance between any two points in a plane forms the hypotenuse of a right triangle with sides parallel to the axes. The distances along the axes are the two other sides of the right triangle. Remember, this theorem only holds true for right-angled triangles, which is a frequent condition encountered in coordinate geometry.