Problem 34 - Precalculus with Limits

Question

In Exercises 27-38, find the distance between the points. \( (1, 3) \), \( (3, -2) \)

Step-by-Step Solution

Answer

The distance between the points (1, 3) and (3, -2) is \(\sqrt{29}\)

Step 1: Determine the coordinates

Identify the coordinates for the points. Let (1, 3) be \(x_1, y_1\) and (3, -2) be \(x_2, y_2\)

Step 2: Plug the coordinates into the distance formula

Apply the coordinates to the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), which becomes \(\sqrt{(3 - 1)^2 + ((-2) - 3)^2}\)

Step 3: Calculate the distance

Simplify the equation to find the distance between the two points. This evaluates to \(\sqrt{(2)^2 + ((-5)^2)} = \sqrt{4 + 25} = \sqrt{29}\)

Key Concepts

coordinate geometry

Coordinate geometry helps describe the position of points in a plane using pairs of numbers known as coordinates. Each point is determined by two numbers, usually written in brackets as \(x, y\).

In this exercise, the point \(1, 3\) represents the first location, where \((1)\) is the x-coordinate (horizontal axis) and \(3\) is the y-coordinate (vertical axis). The second point, \(3, -2\), also follows this pattern, with \((3)\) being its x-coordinate and \((-2)\) its y-coordinate.

Understanding coordinate geometry allows us to visualize geometric figures numerically. It establishes relationships by mapping points to numbers. We treat these numerical points similarly to how we would with numbers on a graph grid. This way, calculations like determining distances or finding slopes become more straightforward by working with numerical values.

calculating distance

Calculating the distance between two points on a plane involves using a simple formula derived from coordinate geometry. The distance formula is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), which finds the shortest path between two points.

In our example:

• The first point is labeled as \(x_1\) and \(y_1\) being \(1\) and \(3\), respectively.
• The second point has coordinates \(x_2\) and \(y_2\) as \(3\) and \(-2\).

Plugging these into the formula:

• Subtract the x-coordinates: \(3 - 1 = 2\)
• Subtract the y-coordinates: \(-2 - 3 = -5\)
• Square the results: \(2^2 = 4\) and \((-5)^2 = 25\)

Calculate the result by taking the square root: \(\sqrt{4 + 25} = \sqrt{29}\).

Pythagorean theorem

The Pythagorean Theorem is essential in understanding the distance formula. Originating from geometry, it concerns right triangles. It states that, for a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides: \(a^2 + b^2 = c^2\).

The connection to the distance formula comes from creating an imaginary right triangle with our points in coordinate geometry. Here:

• The horizontal and vertical distances between the points act as the two shorter sides \((a\) and \(b)\).
• The straight line (or distance) connecting the points is like the hypotenuse \(c\).

The formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) is therefore a direct application of \(a^2 + b^2 = c^2\) to find the diagonal distance without needing the triangle to be visible. This is why the distance formula is quick and effective – it's based on one of geometry's most enduring principles.