Problem 22 - Algebra 1

Question

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(-2,-1),(3,-3)$$

Step-by-Step Solution

Answer

The distance between the points (-2,-1) and (3,-3) is approximately 5.39 when rounded to the nearest hundredth

Step 1: Identify the coordinates of the points

The points are (-2,-1) and (3,-3). Therefore, the coordinates for point 1 (x_1, y_1) are (-2,-1) and for point 2 (x_2, y_2) are (3,-3).

Step 2: Apply the distance formula

Input the coordinates of the two points into the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 3: Substitute the coordinates into the formula

Substitute the coordinates into the formula: \(\sqrt{(3 - (-2))^2 + ((-3) - (-1))^2}\).

Step 4: Simplify the expression

Simplify the expression inside the square root: \(\sqrt{(5)^2 + ((-2))^2)} = \sqrt{25 + 4} = \sqrt{29}\).

Step 5: Round to nearest hundredth

If necessary, round the result to the nearest hundredth: \(\sqrt{29}\) approximately equals 5.39. So the distance between the points is 5.39 when rounded to the nearest hundredth.

Key Concepts

Coordinate Plane

Understanding the coordinate plane is crucial to solve various geometric problems, including finding the distance between two points. The coordinate plane is a two-dimensional surface formed by the intersection of two lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. These axes divide the plane into four quadrants, each with a specific sign for the coordinates.

Points on this plane are represented by pairs of numbers, known as coordinates \( (x, y) \). The first number, or x-coordinate, signifies how far along the x-axis the point is, whereas the y-coordinate indicates the point's position along the y-axis. For example, the point \( (-2, -1) \) lies in the third quadrant, where both x and y values are negative.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. Stated simply, in a right triangle, the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\): \[c^2 = a^2 + b^2\].

This theorem is the foundation of the distance formula, as it allows us to treat the distance between any two points on the coordinate plane as the hypotenuse of a right triangle with sides parallel to the axes. By finding the lengths of these sides (the differences in x and y coordinates), we can calculate the distance using this theorem.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators that represent a specific value. In the context of the distance formula, we use algebraic expressions to symbolize the differences between the coordinates of two points. For instance, the expression \(x_2 - x_1\) represents the horizontal difference, while \(y_2 - y_1\) represents the vertical difference.

When we calculate the distance between two points, we first establish the algebraic expressions for these differences and then square them, which eliminates any negative values that would complicate the calculation. The sum of these squares then forms a new algebraic expression inside the square root, which symbolizes the distance between the original points.

Square Roots

Square roots are mathematical operations that answer the question 'What number, when multiplied by itself, gives the original number?'. The square root of a number 'n' is written as \(\sqrt{n}\). Taking the square root is the inverse operation of squaring.

In our problem, after squaring and summing the differences between the point coordinates, we end up with an expression under a square root sign. This is the last step in applying the distance formula, as taking the square root yields the actual distance between the points. It's often necessary to round the result of square roots to a sensible level of precision—in our example, to the nearest hundredth—since exact square roots are rare for non-square numbers.