Problem 45 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((-2,-8),(3,4)\)

Step-by-Step Solution

Answer

The exact distance between the points \((-2,-8)\) and \((3,4)\) is 13.

Step 1: Understand the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula derives from the Pythagorean theorem to find the hypotenuse of a right triangle.

Step 2: Identify the Coordinates

Identify the coordinates of the two points. In this case, the points are \((-2, -8)\) and \((3, 4)\). Here, \((x_1, y_1) = (-2, -8)\) and \((x_2, y_2) = (3, 4)\).

Step 3: Substitute into the Distance Formula

Substitute the coordinates into the distance formula: \[ d = \sqrt{(3 - (-2))^2 + (4 - (-8))^2} \] Simplifying inside the parentheses gives: \[ d = \sqrt{(3 + 2)^2 + (4 + 8)^2} \] \[ d = \sqrt{5^2 + 12^2} \]

Step 4: Calculate the Squares

Now calculate the squares of the differences: \[ 5^2 = 25 \quad \text{and} \quad 12^2 = 144 \] Substitute into the equation: \[ d = \sqrt{25 + 144} \] \[ d = \sqrt{169} \]

Step 5: Simplify the Square Root

Finally, simplify the square root: \[ \sqrt{169} = 13 \] Thus, the exact distance between the points \((-2, -8)\) and \((3, 4)\) is 13.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics, particularly in geometry. It describes the relationship between the sides of a right triangle. When you have a triangle with one 90-degree angle, the square of the hypotenuse (the side opposite the right angle), is equal to the sum of the squares of the other two sides. The theorem is expressed mathematically as:\[ a^2 + b^2 = c^2 \]where:

• \( a \) and \( b \) are the lengths of the shorter sides of the triangle (also known as the "legs"), and
• \( c \) is the length of the hypotenuse.

This theorem forms the foundation for the distance formula used in the coordinate plane. By viewing two points and their connecting lines as creating a right triangle, we apply the Pythagorean Theorem to compute the exact distance between them. The horizontal and vertical distances become the two legs of the triangle, and the line connecting the two points is treated as the hypotenuse.

Coordinate Plane

The coordinate plane, often referred to as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It consists of two perpendicular axes:

• The horizontal axis is known as the \( x \)-axis.
• The vertical axis is known as the \( y \)-axis.

These axes intersect at a point called the origin, designated as (0,0). Every point on the plane is represented as an ordered pair \((x, y)\), detailing its position relative to the origin.

When we plot two separate points on this plane, they can be connected to form a straight line, visualized as the hypotenuse of a right triangle. The difference in their \( x \) values gives the base of the triangle, while the difference in their \( y \) values gives the height. Understanding this allows for the application of the distance formula to find the precise measure between points as in the given exercise.

Square Root

A square root is a value that, when multiplied by itself, yields the original number. The square root function is denoted by the symbol \( \sqrt{} \) and is critical in simplifying equations involving squares, like those in the expressions derived from the Pythagorean Theorem. For example, the square root of 25 is 5, because \( 5 \times 5 = 25 \).

So, when you calculate the distance between two points, you begin by determining squares, add them together, and then take the square root to find the length of the line segment connecting them. This step simplifies the value to its actual distance measure, as demonstrated in the exercise.

Simplifying a square root means finding a simpler or more familiar form of a number. In our case, we discovered that the square root of 169 is 13, hence confirming the calculation of distance. Knowledge of handling square roots is essential for accuracy in both basic and advanced mathematics, especially when dealing with quadratic functions and geometric interpretations.