Problem 32 - Algebra and Trigonometry

Question

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((2,13),(7,1)\)

Step-by-Step Solution

Answer

(a) Points plotted at (2, 13) and (7, 1). (b) Distance is 13 units. (c) Midpoint is (4.5, 7).

Step 1: Plotting the Points

First, we need to plot the points on a coordinate plane. The first point is (2, 13) and the second point is (7, 1). Locate point (2, 13) by moving 2 units along the x-axis and 13 units along the y-axis. Similarly, locate point (7, 1) by moving 7 units along the x-axis and 1 unit along the y-axis.

Step 2: Finding the Distance

Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute the coordinates: \(d = \sqrt{(7 - 2)^2 + (1 - 13)^2}\). Calculate the squared differences: \((7 - 2)^2 = 5^2 = 25\) and \((1 - 13)^2 = (-12)^2 = 144\). Add them together: \(25 + 144 = 169\). Finally, find the square root: \(d = \sqrt{169} = 13\). Therefore, the distance is 13 units.

Step 3: Finding the Midpoint

Use the midpoint formula: \((x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Substitute the coordinates:\((x_m, y_m) = \left( \frac{2 + 7}{2}, \frac{13 + 1}{2} \right)\). Calculate the averages:\((x_m, y_m) = \left( \frac{9}{2}, \frac{14}{2} \right)\). Therefore, the midpoint is \((4.5, 7)\).

Key Concepts

Coordinate Geometry: Plotting Points on a Coordinate Plane

To grasp coordinate geometry, it is essential to understand how to plot points on a coordinate plane. The coordinate plane is a two-dimensional surface formed by two intersecting lines, namely the horizontal x-axis and the vertical y-axis.
The point of intersection of these axes is known as the origin, labeled as (0,0). Each point on this plane is represented by an ordered pair of numbers, known as coordinates.
To plot a point like (2, 13), you move 2 units along the x-axis (horizontal) and then move 13 units up along the y-axis (vertical). Similarly, to plot the point (7, 1), you start from the origin, move 7 spaces to the right on the x-axis, and then 1 space up on the y-axis. This method is the foundational step in graphically representing algebraic equations and geometric figures.

Computing Distance Using the Distance Formula

Finding the distance between two points on a coordinate plane can be easily achieved with the distance formula. This formula emerges from the Pythagorean theorem. When you have two points, \((x_1, y_1)\) and \((x_2, y_2)\),the distance \(d\) between them is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].
The formula essentially calculates the length of the hypotenuse of a right triangle formed by these points. For the points (2, 13) and (7, 1), we calculate:1. Subtract the x-coordinates: - \(7 - 2 = 5\) - Square it: \(5^2 = 25\)2. Subtract the y-coordinates: - \(1 - 13 = -12\) - Square it: \((-12)^2 = 144\)3. Add these squares: - \(25 + 144 = 169\)4. Find the square root: - \(\sqrt{169} = 13\)
So, the distance is 13 units. Using this method ensures that you measure the exact linear distance between points, no matter their position on the plane.

Finding the Midpoint with the Midpoint Formula

To locate the midpoint of a line segment between two points, use the midpoint formula. This formula identifies the point that is exactly halfway along the segment, balancing it perfectly across its span. This is incredibly helpful for breaking down symmetrical points or constructing bisectors.
The formula for the midpoint \((x_m, y_m)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\].
By plugging in the coordinates of points (2, 13) and (7, 1), we determine:1. Average the x-coordinates: - \(\frac{2 + 7}{2} = \frac{9}{2} = 4.5\)2. Average the y-coordinates: - \(\frac{13 + 1}{2} = \frac{14}{2} = 7\)
Thus, the midpoint is at (4.5, 7). Practicing with the midpoint formula allows you to comfortably segment line segments into equal halves or even create new central structures on the plane.