Problem 7 - 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. $$ (0,8),(6,16) $$

Step-by-Step Solution

Answer

Distance is 10 units; midpoint is (3, 12).

Step 1: Plot the points

To plot the points \((0, 8)\) and \((6, 16)\), begin by drawing the Cartesian coordinate plane consisting of horizontal (x-axis) and vertical (y-axis) lines. Then, locate the first point \((0, 8)\) where the x-coordinate is 0 and the y-coordinate is 8, which will be on the y-axis. Next, locate the second point \((6, 16)\) by moving 6 units along the x-axis and 16 units upward along the y-axis.

Step 2: Use the Distance Formula

To find the distance between the points \((0,8)\) and \((6,16)\), use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Substitute \(x_1 = 0\), \(y_1 = 8\), \(x_2 = 6\), \(y_2 = 16\): \[d = \sqrt{(6 - 0)^2 + (16 - 8)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\]Thus, the distance between the points is 10 units.

Step 3: Find the Midpoint using the Midpoint Formula

To find the midpoint \(M\) of the segment joining the points \((0,8)\) and \((6,16)\), use the midpoint formula: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Substitute the coordinates \(x_1 = 0\), \(y_1 = 8\), \(x_2 = 6\), \(y_2 = 16\):\[M = \left(\frac{0 + 6}{2}, \frac{8 + 16}{2}\right) = \left(\frac{6}{2}, \frac{24}{2}\right) = (3, 12)\]Therefore, the midpoint of the segment is \((3, 12)\).

Key Concepts

Distance Formula

The distance formula is a fundamental tool in coordinate geometry. It allows you to calculate the distance between two points in a plane. This is especially useful when you want to determine how far apart two locations are on a map or grid. The formula is derived from the Pythagorean theorem and is expressed as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Here,

• \(x_1\) and \(y_1\) are the coordinates of the first point.
• \(x_2\) and \(y_2\) are the coordinates of the second point.

By analyzing the differences between the x-coordinates and y-coordinates (\(x_2 - x_1\) and \(y_2 - y_1\)), you essentially form a right triangle where the hypotenuse is the line segment between the two points. When you square these differences, add them together, and then take the square root, you find the length of the hypotenuse, which is the distance you want to calculate. For example, when you have points \((0,8)\) and \((6,16)\), substituting into the formula gives: \[d = \sqrt{(6 - 0)^2 + (16 - 8)^2} = \sqrt{100} = 10\] Thus, the distance is 10 units. This formula is invaluable for precise calculations in various fields like physics, astronomy, and navigation.

Midpoint Formula

The midpoint formula is used to find the exact center, or midpoint, of the line segment connecting two points. This is very useful when you need to find an average position along a line or when dividing segments into equal parts. The formula is straightforward and is given by: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\] Here, the midpoint \(M\) is calculated by averaging the x-coordinates and the y-coordinates of the two given points:

• Add the x-coordinates and divide by 2 to find the x-coordinate of the midpoint.
• Add the y-coordinates and divide by 2 to find the y-coordinate of the midpoint.

Using the points \((0,8)\) and \((6,16)\), the midpoint \((M)\) is calculated as follows: \[M = \left(\frac{0 + 6}{2}, \frac{8 + 16}{2}\right) = (3, 12)\] This means the midpoint of the segment is at the point \((3, 12)\). This concept is employed in various applications such as computer graphics, robotics, and even art, where symmetry and balance are critical.

Plotting Points

Plotting points on a coordinate plane is the foundational skill for coordinate geometry. It involves placing points based on their coordinates, which consist of an x-value (horizontal position) and a y-value (vertical position). A two-dimensional Cartesian coordinate system is used, featuring a horizontal x-axis and a vertical y-axis.
To plot a point, follow these steps:

• Locate the x-coordinate on the x-axis.
• From this position, move vertically to the position of the y-coordinate.

Consider plotting the points \((0,8)\) and \((6,16)\).

• First point \((0,8)\) is placed by starting at the origin, moving 0 units on the x-axis (staying on the y-axis), and then moving 8 units up.
• Second point \((6,16)\) is found by moving 6 units right from the origin (along the x-axis), then 16 units up.

When you plot these points, you physically see their relative positions in space. This is crucial for understanding spatial relationships, solving geometrical problems, and visualizing mathematics and data across diverse disciplines like engineering and economics.