Problem 19 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (9,-2),(12,-14) $$

Step-by-Step Solution

Answer

The distance between the points \((9, -2)\) and \((12, -14)\) is approximately 12.37 units.

Step 1: Identify the Coordinates

First, identify the coordinates of the two points. The first point has coordinates \((x_1, y_1) = (9, -2)\), and the second point has coordinates \((x_2, y_2) = (12, -14)\).

Step 2: Write the Distance Formula

The formula for finding the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 3: Substitute the Values into the Formula

Substitute the coordinates of the given points into the distance formula: \[d = \sqrt{(12 - 9)^2 + (-14 + 2)^2}\]

Step 4: Simplify Inside the Square Root

Calculate the differences and their squares:- \(x_2 - x_1 = 12 - 9 = 3\)- \(y_2 - y_1 = -14 - (-2) = -12\)- \((3)^2 = 9\)- \((-12)^2 = 144\)Thus, the expression inside the square root becomes: \[d = \sqrt{9 + 144}\]

Step 5: Calculate the Distance

Add the two values inside the square root and find the square root:\[d = \sqrt{153}\]Therefore, the distance between the two points is approximately 12.37 units when rounded to two decimal places.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that helps us to understand geometry using an algebraic approach. This method allows for the precise positioning of shapes and figures using the coordinate system, which is a grid used to plot points, lines and curves on a plane. It is made up of two perpendicular lines called axes - the horizontal axis known as the x-axis and the vertical axis known as the y-axis. A point in this plane is identified using an ordered pair of values,

• The x-coordinate (or abscissa) tells you how far to move left or right from the origin, where the axes intersect.
• The y-coordinate (or ordinate) tells you how far to move up or down from the origin.

By using these coordinates, we can calculate the distance between two points, as well as other properties such as the slope between them or the area of figures formed by these points. Understanding coordinate geometry provides a foundation for more complex geometric concepts and is an essential skill in fields ranging from engineering to computer graphics.

Midpoint

The Midpoint is a concept in geometry used to find the center point between two endpoints on a line segment. This point divides the line segment into two equal halves. To find the midpoint, we use the Midpoint Formula, which is derived from averaging the x-coordinates and y-coordinates of the two given points.For any two points

• \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((M)\) can be found as:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

This formula ensures that you find an accurate midpoint between the two points in a coordinate plane. The midpoint is useful in various applications like finding the center of a diagonal in a rectangle or determining the point of equitable division on a path.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics that relates to the sides of a right-angled triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed with the equation:\[ c^2 = a^2 + b^2 \]Here, \(c\) is the hypotenuse, and \(a\) and \(b\) are the triangle's other two sides. This theorem not only applies in geometry when working directly with triangles, but also aids in deriving the Distance Formula used in coordinate geometry by treating the distance between two points as the hypotenuse of a right-angled triangle, formed by the difference in the x-coordinates and y-coordinates. This explains the formula

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Understanding the Pythagorean Theorem is crucial, as it bridges simple triangle properties and more complex analytical geometry concepts.