Problem 30 - 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. \((-1,6),(-1,-3)\)

Step-by-Step Solution

Answer

The distance is 9 units; the midpoint is (-1, 1.5).

Step 1: Plot the Points

To plot the points \((-1, 6)\) and \((-1, -3)\) on the coordinate plane, notice that both points have the same x-coordinate, \(-1\). This means these points are vertically aligned on the graph, with \(y = 6\) for the first point and \(y = -3\) for the second point.

Step 2: Calculate the Vertical Distance

Since the points are vertically aligned, the distance between them is simply the difference in the y-coordinates. Use the formula for the distance: \[ d = |y_2 - y_1| \]Substitute the given y-coordinates \[ d = |-3 - 6| = |-9| = 9 \]Therefore, the distance between the points is 9 units.

Step 3: Determine the Midpoint

To find the midpoint between the points \((-1, 6)\) and \((-1, -3)\), use the midpoint formula:\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]Substitute the given coordinates: \[ M = \left(\frac{-1 + (-1)}{2}, \frac{6 + (-3)}{2}\right) = \left(\frac{-2}{2}, \frac{3}{2}\right) = (-1, 1.5) \]Thus, the midpoint of the segment joining the two points is \((-1, 1.5)\).

Key Concepts

Midpoint

The midpoint of a line segment in coordinate geometry is a crucial concept. It represents the exact middle point between two endpoints on a \(2D\) plane. To calculate this, you'll use the midpoint formula:

• For the coordinates \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \(M\) is given by:

\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]This formula finds the average of the \(x\) and \(y\) coordinates, thus giving you the point located directly in between.

In the exercise, the midpoint between points \((-1, 6)\) and \((-1, -3)\) was found by plugging in the coordinates to get \((-1, 1.5)\). This means that on a graph, the middle point of the line segment connecting these two points is at \(-1\) on the \(x\)-axis and \(1.5\) on the \(y\)-axis. Remember, since both \(x\)-coordinates are \(-1\), the midpoint lies vertically in the middle of the two vertical points.

Distance Formula

Distance between two points on a coordinate plane is measured using the distance formula. It's particularly simple for vertical and horizontal points but can be generalized for any two points. When points share the same \(x\)-coordinate (or \(y\)-coordinate), like in our example, the calculation simplifies to:

• \(d = |y_2 - y_1|\) for vertical alignment.
• \(d = |x_2 - x_1|\) for horizontal alignment.

For a more general case where both \(x\) and \(y\) coordinates differ, the distance formula is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In this exercise, since the points \((-1, 6)\) and \((-1, -3)\) share the same \(x\)-coordinate, we use the simplified formula: \(d = |-3 - 6| = 9\). This means the vertical distance between them is 9 units.

Understanding these formulas allows you to gauge distances between any given pair of points effortlessly.

Coordinate Plane

The coordinate plane, sometimes called the Cartesian plane, is a two-dimensional plane featuring a horizontal line (\(x\)-axis) and a vertical line (\(y\)-axis). Both axes intersect at the origin, \((0, 0)\). The plane is divided into four quadrants which assist in determining the positions of points.

Each point on the plane is described by an ordered pair \((x, y)\), where \(x\) denotes the horizontal distance and \(y\) the vertical distance from the origin. In our example, the points \((-1, 6)\) and \((-1, -3)\) are located on this plane. They share the same \(x\)-coordinate, \(-1\), indicating they are vertically aligned.

• The coordinate plane allows for visualization of geometric concepts, making calculations like distance and midpoint straightforward.
• It enables easy interpretation of data and supports various mathematical analyses.

Understanding how to plot and interpret points on the coordinate plane is foundational for solving problems in coordinate geometry.