Problem 26 - 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. \((-2,5),(10,0)\)

Step-by-Step Solution

Answer

Distance: 13 units, Midpoint: (4, 2.5).

Step 1: Plot the Points

Begin by plotting the points (-2, 5) and (10, 0) on the coordinate plane. Locate the point (-2, 5) by moving 2 units to the left along the x-axis and 5 units up along the y-axis. Locate the point (10, 0) by moving 10 units to the right along the x-axis and 0 units up or down along the y-axis.

Step 2: Apply the Distance Formula

To find the distance between the two points, use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substitute in the coordinates \((x_1, y_1) = (-2, 5)\) and \((x_2, y_2) = (10, 0)\). \[ d = \sqrt{(10 + 2)^2 + (0 - 5)^2} \] Calculate each component: \( (10 + 2)^2 = 144 \) and \( (0 - 5)^2 = 25 \). Thus, \[ d = \sqrt{144 + 25} = \sqrt{169} = 13 \].

Step 3: Calculate the Midpoint

To find the midpoint of the segment that joins the two points, use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]. Substitute the coordinates of the points into the equation: \[ M = \left( \frac{-2 + 10}{2}, \frac{5 + 0}{2} \right) \] Simplify each fraction: \( \frac{-2 + 10}{2} = \frac{8}{2} = 4 \) and \( \frac{5}{2} = 2.5 \). Thus, the midpoint is \( (4, 2.5) \).

Key Concepts

Midpoint Formula

The midpoint of a segment is the point that is exactly halfway between the two endpoints on a line. This can be calculated easily using the midpoint formula, which is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]To apply this, simply add the x-coordinates of the two points together and divide by 2 to find the x-coordinate of the midpoint. Do the same with the y-coordinates to find the midpoint's y-coordinate.

• For example, for points (-2, 5) and (10, 0), the midpoint \[ M = \left( \frac{-2 + 10}{2}, \frac{5 + 0}{2} \right) = (4, 2.5) \].
• This means the point (4, 2.5) is equidistant from both (-2, 5) and (10, 0).

This tool is especially helpful in geometry and physics when determining balance or center points.

Coordinate Plane

A coordinate plane is essentially a two-dimensional space where points can be defined using pairs of numbers known as coordinates. This is set up using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
In this grid, any point's location is determined by an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance from the origin.

• The origin (0,0) is where these axes intersect, serving as a reference point for locating all other points.
• The four sections created by these axes are called quadrants.
• Knowing how to navigate the coordinate plane is crucial for plotting and interpreting graphical data.

This understanding forms the basis for graphing points, lines, and curves in algebra and geometry.

Plotting Points

Plotting points on the coordinate plane is the process of locating and marking them by using their coordinates. For each point, you begin at the origin and:

• Move horizontally to the appropriate position based on the x-coordinate. Positive values move right; negative values move left.
• From that x-position, move vertically according to the y-coordinate. Positive values move up; negative values move down.

For instance, to plot the point (-2, 5):

• Start at the origin and move 2 units to the left (since \(-2\) is negative).
• From there, move 5 units up (since \(5\) is positive).

Similarly, to plot (10, 0): move 10 units right from the origin and stop—it remains level since the y-coordinate is zero.
Mastering this simple skill allows you to visualize relationships between various points and lines on the graph.

Distance Between Points

Finding the distance between two points on a coordinate plane is straightforward with the distance formula. Derived from the Pythagorean Theorem, it lets you calculate the straight-line (or Euclidean) distance between any two points with coordinates\((x_1, y_1)\) and \((x_2, y_2)\).
The formula is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• Substitute the coordinates of your points: for (-2, 5) and (10, 0), it becomes \(d = \sqrt{(10 - (-2))^2 + (0 - 5)^2} \).
• This expression simplifies to \(d = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \).

Understanding this method is vital for many real-world applications, including navigation, architecture, and any field requiring precise measurements.