Problem 49 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((-2,1),(3,4)\)

Step-by-Step Solution

Answer

The exact distance between the points is \( \sqrt{34} \).

Step 1: Identify the point coordinates

We are given two points, \((-2,1)\) and \(3,4)\). Let's identify the coordinates of each point. The first point, \((-2,1)\), has coordinates \(x_1 = -2\) and \(y_1 = 1\). The second point, \(3,4)\), has coordinates \(x_2 = 3\) and \(y_2 = 4\).

Step 2: Apply the Distance Formula

We use the distance formula to find the distance between two points \((x_1, y_1)\) and \(x_2, y_2)\). The formula is written as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Plug the coordinates into the formula to find the distance.

Step 3: Calculate the difference of x-coordinates

Subtract the x-coordinate of the first point from the x-coordinate of the second point: \(x_2 - x_1 = 3 - (-2) = 3 + 2 = 5\)

Step 4: Calculate the difference of y-coordinates

Subtract the y-coordinate of the first point from the y-coordinate of the second point: \(y_2 - y_1 = 4 - 1 = 3\)

Step 5: Compute the squares of the differences

Now, square the differences found in the previous steps: \[(x_2 - x_1)^2 = 5^2 = 25\] \[(y_2 - y_1)^2 = 3^2 = 9\]

Step 6: Add the squares of the differences

Add the squared differences:\[25 + 9 = 34\]

Step 7: Find the square root of the sum

To find the distance, take the square root of 34:\[ d = \sqrt{34} \] It cannot be simplified further into an integer or a simpler radical form.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It connects algebra and geometry by utilizing coordinate planes to illustrate points, lines, and figures. This makes it easier to solve geometric problems using algebraic equations. In a 2D plane, we use an x-axis (horizontal) and a y-axis (vertical) to pinpoint the location of points. For example, in our exercise, the points

• er{-2, 1) and (3,4) represent two locations on this plane.
• erCoordinates allow us to measure various aspects like distance, slope, and area directly from a mathematical equation.

By associating them with numbers, coordinate geometry makes geometric problems easier to work with, using arithmetic calculations.

Distance Between Points

Understanding how to calculate the distance between two points on a coordinate plane is fundamental in geometry. The distance is the length of the straight line that directly connects two points. Imagine you have two points, such as er(-2, 1) and (3,4). The distance between them can be found using the distance formula, which is derived from the Pythagorean theorem. The formula is written as: d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}Simply plug in the coordinates into the formula:

Subtract the x-coordinates: \(3 - (-2) = 5\)

Subtract the y-coordinates: \(4 - 1 = 3\)

Square these differences: \(5^2 = 25\) and \(3^2 = 9\)

Add them together: \(25 + 9 = 34\)

Finally, compute the square root to find the distance: \(d = \sqrt{34}\)

This process ensures that the distance calculation is clear, consistent, and reliable regardless of the specific points involved.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as: c^2 = a^2 + b^2But how does this relate to finding the distance between two points in coordinate geometry? When you have two points, such as er(-2, 1) and (3,4), envision creating a right triangle where:

• The horizontal distance between x-coordinates \((3 - (-2) = 5)\) forms one leg of the triangle.
• The vertical distance \((4 - 1 = 3)\) forms the other leg.

The hypotenuse, in this case, is the direct line between the two points – which is exactly the distance we're solving for using the distance formula. By applying the Pythagorean theorem concept (hypotenuse \(= \sqrt{(a^2 + b^2)}\)), we easily transition into the distance formula to determine that d = \sqrt{34}.This shows how closely coordinate geometry and traditional geometric notions like the Pythagorean theorem are related.