Problem 23 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (4,11),(-5,2) $$

Step-by-Step Solution

Answer

The distance between the points (4,11) and (-5,2) is approximately 12.73 when rounded to the nearest hundredth.

Step 1: Identify the coordinates

Identify the coordinates of the two points. We have point 1 as (4,11) and point 2 as (-5,2). So, \(x_1 = 4\), \(y_1 = 11\), \(x_2 = -5\), and \(y_2 = 2\).

Step 2: Substitute into the distance formula

Substitute these coordinates into the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). It yields \(\sqrt{(-5 - 4)^2 + (2 - 11)^2}\).

Step 3: Solve the expression

Simplify the expression inside the square root. That gets us \(\sqrt{(-9)^2 + (-9)^2}\) or \(\sqrt{81 + 81}\).

Step 4: Find the distance

Use the simple square root calculations to find the distance. So, the distance is \(\sqrt{162}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a link between geometry and algebra through graphs involving coordinates and equations. It allows us to solve geometric problems by using algebraic equations, making it a key concept in mathematics, especially when dealing with graphs and shapes.

When you plot points on a coordinate plane, you are working with a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on this plane is defined by an ordered pair (x, y), where 'x' represents the horizontal distance from the origin and 'y' represents the vertical distance from the origin. Determining the distance between two points involves creating a right-angled triangle and using the lengths of its sides to find the hypotenuse, which represents the distance.

Pythagorean Theorem

The Pythagorean theorem is a foundational principle in geometry that relates the lengths of the sides of a right-angled triangle. The theorem 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 lengths of the other two sides.

This theorem, represented by the equation \( a^2 + b^2 = c^2 \), is widely used to calculate the distance between two points in coordinate geometry. When we calculate this distance, the difference in the x-coordinates (\(x_2 - x_1\)) and the difference in the y-coordinates (\(y_2 - y_1\)) form the legs of the right triangle, and the distance between the points is the hypotenuse (c). The distance formula is derived directly from the Pythagorean theorem.

Square Root

The square root operation is essential when working with the Pythagorean theorem, particularly when finding the distance between two points. The square root of a number is a value that, when multiplied by itself, gives the original number. For the equation \(a^2 = b\), 'a' is the square root of 'b'.

In the context of the Pythagorean theorem, once we have the sum of the squares of the legs of a right triangle, we find the length of the hypotenuse by taking the square root of this sum. This final step yields the actual distance between our two points in coordinate geometry. As in our example, after calculating \(81 + 81\), we determine the distance by computing \(\sqrt{162}\), which simplifies to the actual distance between the points.