Problem 5 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the coordinates

First, identify the coordinates of the two points. In this exercise, the two points provided are (0,0) and (-3,4). So, x₁=0, y₁=0, x₂=-3 and y₂=4.

Step 2: Substitute the values into the formula

Next, substitute these values into the distance formula. √[((-3)-0)² + ((4)-0)²] = √[(-3)² + 4²] = √[9 + 16]

Step 3: Calculate the value

Then, calculate the value under the square root and take the square root of it to find the distance. The square root of 25 is 5.

Key Concepts

Coordinate Geometry

When solving problems in algebra, particularly those involving finding distances, coordinate geometry is an invaluable tool. This area of mathematics allows us to visually interpret and solve algebraic problems by placing them on a graph with two axes: the x-axis (horizontal) and the y-axis (vertical). Each point in the system is represented by an ordered pair (x, y), which denotes its position relative to these two axes.

The concept of coordinate geometry becomes especially useful when dealing with shapes, lines, and distances in a two-dimensional plane. By plotting points on this grid, we can then apply algebraic formulas to calculate lengths and areas, or even to understand the relationships between different geometric entities. For students to master problems in coordinate geometry, getting comfortable with plotting points and visualizing the space between them is a crucial first step.

Pythagorean Theorem

One of the most celebrated principles in geometry is the Pythagorean Theorem, a formula that relates the lengths of the sides of a right triangle in a beautifully simple way. This theorem states that for 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. Mathematically, this is described as:\[ a^2 + b^2 = c^2 \]
where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the triangle's other two sides. The theorem can be used in a multitude of scenarios, including our context of finding the distance between two points on a coordinate plane. By thinking of this distance as the hypotenuse and the differences in x and y coordinates as the other two sides of a right triangle, we can apply the Pythagorean Theorem directly to find the length we're seeking.

Point Distance Calculation

In the realm of algebra, calculating the distance between two points on a coordinate grid is a straightforward application of the Pythagorean Theorem. This application is encapsulated in the distance formula, which is derived directly from the theorem. To determine the distance 'd' between two points (x₁, y₁) and (x₂, y₂), the formula is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
To apply this formula correctly, as we did in our example, it's paramount to work step by step: identify the coordinates, plug them into the formula, and perform the arithmetic. This ensures precision and helps to avoid common mistakes like mixing up coordinates or squaring negative numbers incorrectly. Practice with this formula builds a deeper understanding and enhances students' ability to approach more complex algebraic problems involving distances.