Problem 5 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(0,0)\( and \)(-3,4)$$

Step-by-Step Solution

Answer

The distance between the points (0,0) and (-3,4) is 5 units.

Step 1: Understand and Apply the Distance Formula

The distance formula is derived from Pythagoras' theorem and is used to measure the distance between two points in a 2D space. The formula is \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). In this case, our points are (0,0) and (-3,4). The first point will be considered as (\(x_1,y_1\)) and the second point will be considered as (\(x_2,y_2\)). Now insert these values into the formula.

Step 2: Substitute the values into the formula

Let's substitute the given points coordinates into the formula. So, \(d = \sqrt{((-3)-0)^2 + (4-0)^2} = \sqrt{(-3)^2 + 4^2}\)

Step 3: Calculate the Squares

Simplify the equation further by calculating the squares of the values: \(d = \sqrt{9 + 16}\)

Step 4: Add and Take the Square Root

Add the two numbers inside the square root and calculate the square root: \(d = \sqrt{25} = 5\)

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics, more specifically in geometry. It states that in a right-angled 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 can be expressed in the equation: \[ a^2 + b^2 = c^2 \] where \( c \) is the length of the hypotenuse, while \( a \) and \( b \) are the lengths of the triangle's other two sides.

When it comes to finding distances between two points on a coordinate plane, we actually form a right-angled triangle with these points and the axes. The distance formula derives from the Pythagorean theorem, where the legs of the triangle represent the difference in the x-coordinates and y-coordinates, and the hypotenuse is the distance between the two points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method allows us to describe geometrical shapes in numerical terms and extract useful information. For example, the position of any point on a two-dimensional plane can be represented by two coordinates: \( x, y \).

With regard to finding distance, we can picture any two points on this plane as the corners of a rectangle, and the distance between them as the diagonal. The difference between their x-coordinates and y-coordinates give us the lengths of the sides of the rectangle. Using these values in the distance formula - which is an application of the Pythagorean theorem in coordinate geometry - allows us to compute the length of the diagonal, i.e., the distance between the points.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. It is represented by the radical symbol \( \sqrt{} \). Square roots are crucial when working with the Pythagorean theorem in the context of the distance formula because after we compute the sum of the squares of the differences in x and y coordinates, we need to take the square root of this sum to find the actual distance between the two points.

In our exercise, after adding the squares of the differences \( (-3)^2 + 4^2 = 9 + 16 = 25 \), we took the square root of 25, which is 5, to find the distance between the points (0,0) and (-3,4). It is also important to note that every positive real number has two square roots, one positive and one negative. However, in the context of distance, we only consider the positive root because distance cannot be negative.