Problem 15 - Algebra 1

Question

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(2,-8),(-3,3)$$

Step-by-Step Solution

Answer

The distance between the points (2, -8) and (-3, 3) is approximately 12.08 (rounded to the nearest hundredth).

Step 1: Identify the coordinates

Here, (x1, y1) = (2, -8) and (x2, y2) = (-3, 3)

Step 2: Apply the distance formula

The formula for distance between two points \((x1, y1)\) and \((x2, y2)\) in a two-dimensional space is \[ \sqrt{(x2-x1)^2 + (y2-y1)^2} \]

Step 3: Substitute the given values in the formula

Replacing the given values we have \[ \sqrt{(-3-2)^2 + (3-(-8))^2} \]

Step 4: Simplify the equation

Simplifying what is inside the square root gives us \[ \sqrt{(-5)^2 + (11)^2} \], which further simplifies to \[ \sqrt{25 + 121} \]

Step 5: Compute the square root

Calculating the square root of the sum, we get \[ \sqrt{146} \]

Step 6: Round to the nearest hundredth

Square root of 146 is approximately 12.08 (rounded to the nearest hundredth).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a coordinate system to study geometric figures. This system helps us define the location of points in a plane using ordered pairs, known as coordinates. These coordinates are generally represented as

• the x-coordinate (horizontal distance from the origin)
• the y-coordinate (vertical distance from the origin)

Understanding coordinate geometry is essential for precisely determining the positions of points and calculating distances between them. In our exercise, the given coordinates are the foundation for finding the distance between two points. We have two points: (2, -8) and (-3, 3). The numbers show us where each point is located on the x-axis and y-axis. In simpler terms, coordinate geometry allows us to map out a path to solve problems involving distances across a grid or plane.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry, linking the sides of a right triangle. It states that in a right-angled 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. This relationship is expressed by the formula:\[ a^2 + b^2 = c^2 \]where:

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

The distance formula used for finding the distance between two points in coordinate geometry is derived from this theorem. When we apply the distance formula, \[ \sqrt{(x2-x1)^2 + (y2-y1)^2} \]we are essentially interpreting the two point coordinates as forming a right triangle. By doing this, the legs of the triangle are the differences in the x and y coordinates, and the distance we want to find is analogous to the hypotenuse. This elegant connection between the theorem and our coordinate system lets us precisely measure the straight-line distance between any two points.

Distance Calculation

Distance calculation is a method used to find how far apart two points are in a plane. Using the coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \), we calculate the distance by using the distance formula:\[ \sqrt{(x2-x1)^2 + (y2-y1)^2} \]In our example, we substitute:

• \( (x_1, y_1) = (2, -8) \)
• \( (x_2, y_2) = (-3, 3) \)

Putting these into the formula gives:\[ \sqrt{(-3-2)^2 + (3-(-8))^2} \]Simplifying inside the square root, we solve:

• \( (-5)^2 = 25 \)
• \( (11)^2 = 121 \)

Adding these results gives 146, so our final distance is:\[ \sqrt{146} \]When approximating, this results in about 12.08 when rounded to the nearest hundredth. Understanding this calculation allows us to determine how direct or straightforward a path is between two coordinates on a plane.