Problem 29 - 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. \((6,-2),(-1,3)\)

Step-by-Step Solution

Answer

Plot the points, distance is \(\sqrt{74}\), mid-point is \((2.5, 0.5)\).

Step 1: Plotting the Points

To visualize the points, we need to plot them on a standard coordinate plane. The first point, \((6, -2)\), lies on the right of the Y-axis as it has a positive X-value and 2 units below the X-axis since its Y-value is -2. The second point, \((-1, 3)\), is on the left of the Y-axis as it has a negative X-value and 3 units above the X-axis due to its positive Y-value. Plot these two points to start our calculations.

Step 2: Using the Distance Formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Inserting the given points \((6, -2)\) and \((-1, 3)\), we have:\[d = \sqrt{((-1) - 6)^2 + (3 - (-2))^2} = \sqrt{(-7)^2 + (5)^2} = \sqrt{49 + 25} = \sqrt{74}\].So, the distance between the two points is \(\sqrt{74}\).

Step 3: Finding the Mid-point

The formula for finding the mid-point of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Substituting the given points \((6, -2)\) and \((-1, 3)\):\[M = \left(\frac{6 + (-1)}{2}, \frac{-2 + 3}{2}\right) = \left(\frac{5}{2}, \frac{1}{2}\right)\].Thus, the mid-point is \((2.5, 0.5)\).

Key Concepts

Distance Formula

The distance formula is a fundamental tool in coordinate geometry used to find the distance between two points on a coordinate plane. It is derived from the Pythagorean theorem, which relates to the sides of a right-angled triangle.
This formula calculates the straight-line distance between two points \(x_1, y_1\) and \(x_2, y_2\) and is given by:

• \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

To apply this formula, you substitute the x and y coordinates of your points into the equation. This helps in determining the distance no matter where the points lie on the plane.
For example, with points \(6, -2\) and \(-1, 3\), we plugged their coordinates into the formula and found:

• \[d = \sqrt{((-1) - 6)^2 + (3 - (-2))^2} = \sqrt{49 + 25} = \sqrt{74}\]

So, the calculated distance turns out to be \(\sqrt{74}\), which can be further simplified if necessary.

Midpoint Formula

The Midpoint Formula is used to find the point that is equidistant from two given points on a coordinate plane. This point, called the midpoint, represents the average position between two points.
The formula for the midpoint \(M\) of a line segment connecting two points \(x_1, y_1\) and \(x_2, y_2\) is:

• \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

This formula essentially calculates the average of the x-coordinates and the y-coordinates separately to determine the midpoint coordinates.
In our example with points \(6, -2\) and \(-1, 3\), we used the formula:

• \[M = \left(\frac{6 + (-1)}{2}, \frac{-2 + 3}{2}\right) = \left(\frac{5}{2}, \frac{1}{2}\right)\]

Thus, the midpoint between these two points is \(2.5, 0.5\), located centrally on the line joining them.

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is defined by two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical).
Each point on the coordinate plane is defined by an ordered pair \(x, y\), where the x-value specifies the horizontal position and the y-value specifies the vertical position. This system of coordinates enables precise placement and measurement of points.
The coordinate plane is divided into four quadrants:

• Quadrant I, where both x and y values are positive.
• Quadrant II, where x values are negative and y values are positive.
• Quadrant III, where both x and y values are negative.
• Quadrant IV, where x values are positive and y values are negative.

When plotting points like \(6, -2\) and \(-1, 3\), understanding the coordinate plane helps us see their exact positions relative to the axes. The first point lies to the right and below the origin, whereas the second point is left and above the origin.