Problem 9 - 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. $$ (-3,-6),(4,18) $$

Step-by-Step Solution

Answer

Distance: 25 units, Midpoint: \(\left(\frac{1}{2}, 6\right)\).

Step 1: Plot the Points

Begin by identifying the coordinates of each point. The points given are \((-3, -6)\) and \((4, 18)\). Plot these points on a coordinate plane with the x-axis representing the first coordinate and the y-axis representing the second coordinate. The first point is located 3 units to the left and 6 units down from the origin, and the second point is located 4 units to the right and 18 units up from the origin.

Step 2: Calculate the Distance Between the Points

Use the distance formula to find the distance between two points in a coordinate plane, which is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute the given points into the formula: \((-3, -6)\) and \((4, 18)\). Thus, \(d = \sqrt{(4 + 3)^2 + (18 + 6)^2} = \sqrt{7^2 + 24^2}\). Calculate and simplify: \(d = \sqrt{49 + 576} = \sqrt{625} = 25\). The distance is 25 units.

Step 3: Find the Midpoint of the Segment

Use the midpoint formula to find the midpoint of the segment connecting the two points, which is \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute the points \((-3, -6)\) and \((4, 18)\) into the formula: \(M = \left(\frac{-3 + 4}{2}, \frac{-6 + 18}{2}\right) = \left(\frac{1}{2}, 6\right)\). Therefore, the midpoint is \(\left(\frac{1}{2}, 6\right)\).

Key Concepts

Distance Formula

The distance formula is a crucial tool in coordinate geometry for finding the distance between two points on a plane. To use it effectively, it is essential to understand its derivation and application. The formula is given by:

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

This formula is derived from the Pythagorean theorem. Think of the line segment between the two points as the hypotenuse of a right triangle. The difference in the x-coordinates \((x_2 - x_1)\) and the difference in the y-coordinates \((y_2 - y_1)\) are the two legs of this hypothetical right triangle. Thus, when you square these differences, add them, and then take the square root, you get the length of the hypotenuse, or the distance between the two points.
In our exercise, the distance between the points \((-3, -6)\) and \((4, 18)\) can be found by substituting these coordinates into the formula. The calculation yields:
\[ d = \sqrt{(4 + 3)^2 + (18 + 6)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]The final result is a distance of 25 units, which represents the straight-line separation between the two points on the coordinate plane.

Midpoint Formula

The midpoint formula is a handy tool in coordinate geometry that helps in finding the exact middle point on a line segment that connects two given points. This midpoint represents the average position between the two coordinates.
The formula for calculating the midpoint \(M\) is:

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

The idea here is to find the mean or average of the x-coordinates and the y-coordinates separately. This gives you the point that is equidistant from each point in the segment.
For example, using the points \((-3, -6)\) and \((4, 18)\), we calculate the midpoint as:

• \( M_x = \frac{-3 + 4}{2} = \frac{1}{2} \)
• \( M_y = \frac{-6 + 18}{2} = 6 \)

Thus, the midpoint of the segment is \(\left( \frac{1}{2}, 6 \right)\).
This midpoint calculation is especially useful in many practical applications such as splitting distances for construction, finding centers in engineering designs, or figuring out meeting points in navigation.

Graphing Points

Graphing points is a fundamental skill in coordinate geometry, allowing you to visually represent algebraic equations and understand spatial relationships between different points. A coordinate plane consists of two axes:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

Each point on this plane is represented by a pair of values \((x, y)\), where \(x\) is the position along the x-axis and \(y\) is the position along the y-axis.
For accurate graphing, follow these steps:

• Locate the x-coordinate on the horizontal axis.
• Locate the y-coordinate on the vertical axis.
• Plot the point where these two positions meet on the grid.

In our problem, we are given the points \((-3, -6)\) and \((4, 18)\). The first point is placed 3 units left and 6 units down from the origin, making it easier to visualize its location. The second point is 4 units right and 18 units up, showcasing the distance on the plane.
Accurate plotting provides better comprehension of problems involving distances and midpoints. It visually supports derived calculations and strengthens the connection between theoretical concepts and their real-world applications in various fields such as physics, graphics design, and more.