Problem 20 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (2,1),(8,4) $$

Step-by-Step Solution

Answer

The distance between the two points (2,1) and (8,4) is approximately 6.71.

Step 1: Identify Coordinates

First determine the given points. The coordinates for Point 1 are (2,1) and the coordinates for Point 2 are (8,4).

Step 2: Apply Distance Formula

Plug the coordinates into the distance formula. Substituting the given values into the distance formula results in \(d = \sqrt{(8 - 2)^2 + (4 - 1)^2}\)

Step 3: Calculate Distance

Evaluate the above expression. Solving it further leads to, \(d = \sqrt{6^2 + 3^2}\) which simplifies to \(d = \sqrt{36 + 9}\) and finally \(d = \sqrt{45}\) resulting in \(d = 6.71\) after rounding to the nearest hundredth as instructed.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating branch of mathematics. It focuses on representing geometric figures using a coordinate system. In simpler terms, it is a way to describe shapes and positions using numbers. By using the coordinate plane, which consists of two axes (x-axis and y-axis), we can pinpoint the exact location of any point with coordinates.

Here are a few key aspects of coordinate geometry:

• **Coordinates**: Coordinates (x, y) help to locate the position of a point on the plane. The first number refers to the horizontal position (x-axis), while the second refers to the vertical position (y-axis).
• **Coordinate Plane**: Divides space into four quadrants, which helps in defining the position of points more specifically.

This mathematical tool is crucial because it allows us to employ algebraic techniques to solve geometric problems, like finding distances or midpoints, which simplifies many complex calculations.

Distance Between Two Points

The distance formula is a handy tool derived from the Pythagorean theorem. It is used to determine the distance between two points on an xy-plane. Understanding this formula makes solving problems involving point locations much easier.

Here's how the distance formula works:

• Given two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) between these points is calculated as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Step-by-step, you first subtract the x-coordinates and square the result.
• Then, subtract the y-coordinates and square that result.
• Add the two squares together.
• Finally, take the square root of the sum to find the distance.

For the exercise given, plugging in the coordinates (2,1) and (8,4) into the formula will calculate the distance between the two points, which ultimately rounds to 6.71. It makes complex-circle-packing and spatial reasoning challenges much easier to navigate.

Mathematics Exercise

Mathematics exercises like the one provided are essential for developing problem-solving skills and understanding fundamental concepts. Practice solidifies knowledge, enabling students to become proficient in mathematical procedures.

These exercises often include:

• **Recognizing Coordinates**: Identifying the numbers associated with each point's position.
• **Applying Formulas**: Using specific formulas like the distance formula to compute solutions.
• **Calculating**: Performing algebraic operations like subtraction and square rooting, which are necessary steps in deriving the final answer.

The given exercise helps students understand and apply the distance formula in a practical context. It reinforces the concept by requiring computation and rounding, thereby offering a complete learning experience. By practicing such exercises, learners enhance their spatial thinking and analytical skills, crucial for further studies in mathematics.