Problem 19 - College Algebra with Modeling and Visualization

Question

Find the exact distance between the two points. Where appropriate, also give approximate results to the nearest hundredth. $$ (-3,2),(-3,10) $$

Step-by-Step Solution

Answer

The exact distance between the points is 8.

Step 1: Identifying coordinates

First, we identify the coordinates of the two given points. The points are \(-3, 2\) and \(-3, 10\). The first point, A, has coordinates \(x_1 = -3\) and \(y_1 = 2\). The second point, B, has coordinates \(x_2 = -3\) and \(y_2 = 10\).

Step 2: Check if the points lie on a vertical line

Since both points have the same x-coordinate \(-3\), they lie on a vertical line. Thus, the formula for distance between the points simplifies to finding the difference in their y-coordinates.

Step 3: Calculate the distance

The distance between two points on a vertical or horizontal line is the absolute difference between their respective y or x coordinates. Therefore, the distance \(d\) is:\[ d = |y_2 - y_1| = |10 - 2| = 8 \].

Step 4: Approximate the distance

Since the calculated distance is a whole number, the approximate result to the nearest hundredth is the same as the exact distance, 8.

Key Concepts

Vertical Line

In geometry, a vertical line is a line that goes straight up and down. It is characterized by being parallel to the y-axis and having a constant x-coordinate. When two points share the same x-coordinate, they lie on a vertical line. This simplifies the calculation of distance between them because one only needs to consider the difference in their y-coordinates. For example, the points \((-3, 2)\) and \((-3, 10)\) lie on a vertical line because they share the same x-coordinate of \(-3\). This means the entire focus shifts to how far apart they are vertically.

Coordinates

Coordinates are a set of values that define the exact position of a point on a plane. In a two-dimensional plane, this is often expressed as \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position. In our example, point \(A\) has coordinates \((-3, 2)\) and point \(B\) has coordinates \((-3, 10)\). These coordinates help us determine the specific position of each point and are crucial for finding the distance between them. By understanding the coordinates of each point, we can quickly assess if they are on a vertical or horizontal line.

Absolute Difference

Absolute difference refers to the positive difference between two numbers. It is calculated by subtracting one number from the other and taking the absolute value of the result. This is often used to find the distance on a number line or between coordinates, regardless of direction. For example, to find the distance between the y-coordinates of the points \((-3, 2)\) and \((-3, 10)\), we calculate \( |y_2 - y_1| = |10 - 2| = 8 \). Since the distance is always a positive number, we use absolute value to ensure our final answer is correct.