Problem 45 - College Algebra with Corequisite Support

Question

For each of the following exercises, use the graph in the figure below. Find the distance that (-3,4) is from the origin.

Step-by-Step Solution

Answer

The distance from (-3,4) to the origin is 5 units.

Step 1: Identify the Coordinates

The point given is (-3, 4). We need to determine how far this point is from the origin (0, 0).

Step 2: Use the Distance Formula

To find the distance from a point (x, y) to the origin (0, 0), use the distance formula \(d = \sqrt{x^2 + y^2}\). For (-3, 4), this becomes\(d = \sqrt{(-3)^2 + 4^2}\).

Step 3: Square the Coordinates

Calculate (-3)^2 and 4^2. The results are 9 and 16, respectively.

Step 4: Add the Squares

Add the squared values: 9 + 16 = 25.

Step 5: Calculate the Square Root

Find the square root of 25, which is 5. This is the distance from (-3, 4) to the origin.

Key Concepts

Coordinates

Coordinates are essential when working with graphs and points. They are a pair of numbers \(x, y\) that describe the exact position of a point on a two-dimensional plane. The first number, \(x\), represents how far left or right the point is from the vertical axis, known as the y-axis. The second number, \(y\), shows how far up or down the point is from the horizontal axis, called the x-axis.

• Positive x-coordinates indicate a position to the right of the y-axis.
• Negative x-coordinates mean a location to the left.
• Positive y-coordinates suggest a position above the x-axis.
• Negative y-coordinates place the point below the x-axis.

Understanding coordinates is fundamental to finding distances between points on a graph.

Square Root

The square root is a significant mathematical concept, especially when dealing with geometry and distances. The square root of a number is a value that, when multiplied by itself, gives the original number.
For example, the square root of 25 is 5, because \(5 \times 5 = 25\).
Using the square root is crucial in finding the distance between two points. It helps simplify the solution to equations like the distance formula. When engaging with the distance formula, you'll often encounter a square root as the last step, as it converts the squared distance back to its original form.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used to find distances in geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The formula is: \[a^2 + b^2 = c^2\]
in which \(c\) represents the hypotenuse, and \(a\) and \(b\) are the other two sides of the triangle. This theorem is the backbone of the distance formula. When you calculate the distance between a point and the origin, you're applying this theorem by treating the coordinate line as a right triangle. The coordinates serve as the perpendicular sides, while the distance is the hypotenuse.

Origin

The origin is a fundamental point in the coordinate system. It is located at the coordinates \(0, 0\) and serves as the starting point for measuring distances on a graph.
At the origin, both the x-coordinate and y-coordinate are zero. It is the intersection point of the x-axis and y-axis, forming the central reference point for all other points in the coordinate plane.
When calculating the distance from the origin to any point, you are essentially measuring how far that point is from this central reference spot. Understanding the role of the origin is crucial for correctly applying the distance formula, as it helps define the position of other points relative to the center of the graph.