Problem 13 - Algebra and Trigonometry

Question

\(7-14\) A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them. $$ (3,4),(-3,-4) $$

Step-by-Step Solution

Answer

(a) Points plotted: (3,4) and (-3,-4). (b) Distance: 10 units. (c) Midpoint: (0,0).

Step 1: Plot the Points

First, locate the points on a coordinate plane. Point \((3, 4)\) is located in the first quadrant, 3 units right and 4 units up from the origin (0,0). Point \((-3, -4)\) is located in the third quadrant, 3 units left and 4 units down from the origin.

Step 2: Use the Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute the coordinates of the points \((3, 4)\) and \((-3, -4)\) into the formula: \[d = \sqrt{(-3 - 3)^2 + (-4 - 4)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\]. Thus, the distance between the points is 10 units.

Step 3: Use the Midpoint Formula

To find the midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Substitute the coordinates into the formula: \[M = \left( \frac{3 + (-3)}{2}, \frac{4 + (-4)}{2} \right) = \left( 0, 0 \right)\]. Thus, the midpoint is \((0,0)\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot points, lines, and curves. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point known as the origin, represented as (0, 0).
The coordinate plane is used to graphically represent algebraic equations and to visualize mathematical concepts in a spatial manner.
Each point on a coordinate plane is identified by an ordered pair of numbers, written as (x, y), where

• x: Represents the horizontal distance from the origin.
• y: Represents the vertical distance from the origin.

In the given exercise, point (3, 4) is plotted by moving 3 units to the right and 4 units up from the origin. Similarly, point (-3, -4) is plotted by moving 3 units to the left and 4 units down.

Midpoint Formula

The midpoint formula is a mathematical tool used to find the exact middle point, or midpoint, of a line segment connecting two points in the coordinate plane. This is particularly useful in geometry for drawing bisectors or finding equidistant points. The midpoint, M, of a segment connecting points

• Point (x₁, y₁) and
• Point (x₂, y₂) is calculated using the formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
This formula averages the x-coordinates and y-coordinates of the two points to find the center.
For the points (3, 4) and (-3, -4), substituting these into the formula gives us: \[ M = \left( \frac{3 + (-3)}{2}, \frac{4 + (-4)}{2} \right) = (0, 0) \]
Thus, in this example, the midpoint is precisely at the origin.

Quadrants

The coordinate plane is divided into four sections called quadrants. These quadrants help in identifying the position of a point relative to the axes, and they are numbered counterclockwise starting from the upper right.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

Identifying in which quadrant a point lies can help in predicting what kind of calculation or graphing adjustments may be needed.
In the exercise, point (3, 4) is located in Quadrant I, as both coordinates are positive. Point (-3, -4) is in Quadrant III, where both coordinates are negative, showcasing the symmetry of the coordinate plane.