Problem 51 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((-1,-6),(3,-4)\)

Step-by-Step Solution

Answer

The exact distance is \(2\sqrt{5}\).

Step 1: Understand the Distance Formula

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) in a coordinate plane is given by the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For the points (-1, -6) and (3, -4), \(x_1 = -1, y_1 = -6, x_2 = 3, y_2 = -4\).

Step 2: Calculate the Differences

Find the differences in the x-coordinates and y-coordinates. For \(x\): \(x_2 - x_1 = 3 - (-1) = 3 + 1 = 4\). For \(y\): \(y_2 - y_1 = -4 - (-6) = -4 + 6 = 2\).

Step 3: Square the Differences

Square the differences you found. \(x\)-difference is 4, so \(4^2 = 16\). \(y\)-difference is 2, so \(2^2 = 4\).

Step 4: Apply the Distance Formula

Plug these squared differences into the distance formula: \(\sqrt{16 + 4} = \sqrt{20}\).

Step 5: Simplify the Square Root

Simplify \(\sqrt{20}\). Since \(20 = 4 \times 5\), we have: \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\).

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental tool in geometry that helps us locate points and understand their relationships. It is essentially a grid formed by two number lines intersecting at right angles. These number lines are commonly known as axes. The horizontal axis is called the x-axis, while the vertical one is the y-axis. Every point on the plane is represented by a pair of numbers written as \(x, y\). These numbers are called coordinates.

When working with distances between points, the coordinate plane becomes invaluable. It allows us to visualize the location of points. For example, the points (-1, -6) and (3, -4) are each located based on how far they are from the origin of the plane (where the axes meet at (0, 0)).

When finding the distance between two points, you essentially move horizontally and vertically along the grid to connect them. Understanding this layout is crucial to using the distance formula effectively.

Square Root Simplification

Simplifying square roots can make calculations much easier, particularly when dealing with distance in geometry. A square root asks what number, multiplied by itself, equals the original number under the radical sign.\[ \quad For example, \sqrt{4} is simplified because 2 x 2 equals 4, so \sqrt{4} = 2. \]

Now, let's consider the square root of a number like \(20\). The factorization process helps here. Break down \(20\) into its smallest components, which are \(4 \times 5\). We know \(4\) is a perfect square, which gives us \(\sqrt{4} \) and equals \(2\).

Hence, \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\). By simplifying square roots this way, we make distance calculations more manageable.

Pythagorean Theorem

The Pythagorean Theorem is a powerful tool in mathematics that connects the sides of a right triangle. In any right triangle, the theorem tells us that the sum of the squares of the two shorter sides equals the square of the longest side, known as the hypotenuse.

When we look at the distance between two points on a coordinate plane, think of creating a right triangle where these points are connected by the hypotenuse. The differences between their x and y coordinates form the other two sides.

• The length of the x-side is \(x_2 - x_1\).
• The length of the y-side is \(y_2 - y_1\).

So to find the direct distance, you square these lengths, sum them up, and then take the square root:
\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula is directly derived from the Pythagorean Theorem. Hence, when you calculate distances on a grid, you're using this ancient, yet still relevant, mathematical principle to ensure accuracy.