Problem 10 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-2) \text { and }(4,3) $$

Step-by-Step Solution

Answer

The distance between the points (0,-2) and (4,3) is approximately 6.40.

Step 1: Identify the coordinates of the points

The first point has coordinates (0,-2), so \(x_1 = 0\) and \(y_1 = -2\). The second point has coordinates (4,3), so \(x_2 = 4\) and \(y_2 = 3\).

Step 2: Apply the distance formula

Substitute the coordinates of the points into the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). This gives \(d = \sqrt{(4-0)^2 + (3-(-2))^2}\).

Step 3: Solve the equation

Solve the equation to find the distance. This gives \(d = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}\).

Step 4: Round the answer to two decimal places

If necessary, round the answer to two decimal places. The square root of 41, to two decimal places, is 6.40.

Key Concepts

Coordinate Geometry

Coordinate geometry is a fascinating branch of mathematics that merges algebra and geometry. It involves understanding how geometric figures behave through the use of a coordinate system. In coordinate geometry, every point on a plane is defined by an ordered pair of numbers, known as coordinates. Each pair consists of an x-coordinate (horizontal position) and a y-coordinate (vertical position). This allows us to place points in a two-dimensional space.

For example, in the exercise given, the first point (0, -2) tells us that the point is directly on the y-axis at y = -2. Similarly, the second point (4, 3) means four units to the right of the origin and three units up.

• Coordinate Plane: An imaginary plane dissected by a horizontal x-axis and a vertical y-axis.
• Coordinates: A set of values that show an exact position.
• Origin: The center of the coordinate plane where the x-axis and y-axis intersect (0,0).

By understanding the figures and calculations made on this grid, you can solve numerous real-world problems with ease.

Distance Between Points

The concept of finding the distance between two points in coordinate geometry is an essential skill. The distance formula derives from the Pythagorean Theorem. It calculates the length of the line segment between two points using their coordinates.

The distance formula is expressed as:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Here:

• \((x_1, y_1)\) are the coordinates of the first point.
• \((x_2, y_2)\) are the coordinates of the second point.
• \(d\) is the distance between the two points.

In our example, the coordinates (0, -2) and (4, 3) substitute into the formula giving us:\[d = \sqrt{(4-0)^2 + (3-(-2))^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}\]By applying this formula, you can find the exact distance between any two points on a plane.

Rounding Decimals

Rounding decimals to a specified number of decimal places is a common practice in mathematics, especially when dealing with irrational numbers, like square roots, which have endless decimal places. By rounding, we simplify these numbers for easier understanding and use.

To round a number to two decimal places, you look at the third decimal digit. If it is five or more, you round the second decimal digit up by one. If it is less than five, the second decimal stays the same.

• Example: The square root of 41 is approximately 6.4031, rounded to 6.40.
• Identify the point to round to (two decimal places here).
• Check the digit just beyond your rounding point.

In our example, the calculated distance was \(\sqrt{41}\) which is approximately 6.403. When rounded to two decimal places, this becomes 6.40, offering a balance between precision and simplicity.