Problem 34 - 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. \((5,0),(0,6)\)

Step-by-Step Solution

Answer

Plot the points, the distance is \(\sqrt{61}\), midpoint is \((2.5, 3)\).

Step 1: Understand the Points

The two points given are \((5,0)\) and \((0,6)\). These represent coordinates on the Cartesian plane, where the first number is the x-coordinate, and the second number is the y-coordinate.

Step 2: Plot the Points

Plot the point \((5,0)\) on the x-axis where x=5 and y=0. Plot the point \((0,6)\) on the y-axis where x=0 and y=6. This creates a right triangle with the origin.

Step 3: Use the Distance Formula

To find the distance between the points \((x_1, y_1)\) and \((x_2, y_2)\), use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Substitute the given points: \[d = \sqrt{(0 - 5)^2 + (6 - 0)^2} = \sqrt{(-5)^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}\]. Thus, the distance is \(\sqrt{61}\).

Step 4: Apply the Midpoint Formula

The midpoint \((x_m, y_m)\) of a segment is found using the midpoint formula: \[x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}\]. Substitute the given points: \[x_m = \frac{5 + 0}{2} = 2.5, \quad y_m = \frac{0 + 6}{2} = 3\]. Thus, the midpoint is \((2.5, 3)\).

Key Concepts

Distance Formula

The distance formula is a crucial tool in coordinate geometry, used to determine the length of a line segment between two points on the Cartesian plane.
It helps us quantify and understand the space between points in a plane.
The formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. It allows us to find distances using coordinates.
For two points

• \( (x_1, y_1) \) and \( (x_2, y_2) \)


• The distance \( d \) between them is


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

By subtracting the x-coordinates and y-coordinates separately, squaring them, adding them up, and finally taking the square root, we get the actual distance.
This formula not only simplifies the task of finding distances but also helps in understanding geometric relationships between points.
In the exercise, the points

• (5,0) and (0,6)


• We calculated: \[ \sqrt{(-5)^2 + 6^2} = \sqrt{61} \]


• This shows the distance between the two points to be \( \sqrt{61} \)

It's a handy formula to have at your fingertips when working with the Cartesian plane.

Midpoint Formula

The midpoint formula gives us the center point of a line segment connecting two coordinates. It's essential for bisecting a line and understanding symmetries.
If you know the coordinates of two points, finding their midpoint provides valuable information about the line segment that connects them.
For points

• \( (x_1, y_1) \) and \( (x_2, y_2) \)


• The midpoint \( (x_m, y_m) \) is calculated as:


• \[ x_m = \frac{x_1 + x_2}{2}, \ y_m = \frac{y_1 + y_2}{2} \]

This formula averages the x- and y-coordinates separately, resulting in the midpoint's coordinates.
By finding the midpoint, you can divide a segment into two equal parts or find the line's central point easily.
It can be particularly useful for geometry problems involving bisectors or finding centroid positions.
In the given problem, for points

• (5,0) and (0,6)


• ...we got the midpoint as \( (2.5, 3) \)

This midpoint serves as the center of the segment with equal lengths extending in either direction toward the endpoints.

Cartesian Plane

The Cartesian plane is the foundation of coordinate geometry, consisting of two number lines intersecting at right angles, creating a grid where we can plot points.
It's composed of an x-axis (horizontal line) and a y-axis (vertical line). The intersection point is called the origin, marked as \( (0,0) \).
Every point on this plane can be described using ordered pairs or coordinates, such as \( (x, y) \).

• The first number indicates the position along the x-axis,


• The second number marks the position along the y-axis.

This system allows for easy visualization and analysis of mathematical concepts involving points, lines, and curves.
It's a fundamental aspect of graphing equations and inequalities, enabling us to represent algebraic relationships visually.
In the exercise,

• points \( (5,0) \) and \( (0,6) \) were plotted on this plane.


• Point \((5,0)\) is positioned 5 units along the x-axis and 0 units along the y-axis.


• Point \((0,6)\) is 0 units along the x-axis and 6 units along the y-axis.

Understanding how to navigate and interpret the Cartesian plane is essential for tackling any problem in coordinate geometry.