Problem 21 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (2,1),(-4,16) $$

Step-by-Step Solution

Answer

The distance between the points (2,1) and (-4,16) is approximately 16.15.

Step 1 Identify the points

Identify the points given as \((2,1)\) and \((-4,16)\). Here, (2,1) corresponds to \(x_1 = 2\) and \(y_1 = 1\), and (-4,16) corresponds to \(x_2 = -4\) and \(y_2 = 16\).

Step 2 Substitute in the distance formula

Substitute these values into the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). The calculation becomes \(\sqrt{((-4 - 2)^2 + (16 - 1)^2)}\).

Step 3 Simplify the squares

Simplify the squares in the formula: \(\sqrt{(-6)^2 + (15)^2} = \sqrt{36 + 225}\).

Step 4 Calculate the distance

Calculate the sum under the square root and then take the square root: \(\sqrt{261} ≈ 16.15\).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that combines algebra and geometry to study points, lines, and shapes in a coordinate plane. Each point in this plane is represented by an ordered pair of numbers, often written as \((x, y)\). This is essential for understanding distances between points.

In the distance formula problem, we have been given two points \((2,1)\) and \((-4,16)\). Here’s a quick breakdown of coordinate geometry concepts involved:

• Coordinates: Each point is given as \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.
• Origin: The center of the coordinate plane is \((0, 0)\), where the x-axis and y-axis intersect.
• Distance: Finding the distance between two points involves calculating the straight line that connects them directly within this plane.

Coordinate geometry allows us to visualize algebraic equations and understand geometric concepts using algebraic expressions. By applying the distance formula, we move from these visual representations to precise calculations.

The Role of the Square Root

A square root is a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. In the context of the distance formula, the square root is used to find the actual distance between two points.

When calculating the distance, you reach the step where you sum the squares of differences in both x and y coordinates. This will look like:

• First, for our points, we have \((-4 - 2)^2 + (16 - 1)^2\).
• This simplifies to \(36 + 225 = 261\).

Now, you must take the square root of 261 to find the distance:

• The square root of 261 is approximately \(16.15\).

This step transitions us from the combined squares to the actual straight line distance. Without the square root, we'd be left with a squared value that doesn't represent real-world measurements.

Importance of Squared Differences

The concept of squared differences is crucial in the distance formula as it ensures all distance contributions from each axis are positive.

When finding the distance, the formula squares the difference between x-coordinates and y-coordinates:

• For x–coordinates: \((x_2 - x_1)^2\), or specifically for our problem \((-4 - 2)^2\) becomes \((-6)^2 = 36\).
• For y-coordinates: \((y_2 - y_1)^2\), or for our problem \((16 - 1)^2 = 15^2 = 225\).

The importance of squaring these differences is:

• Positive Values: Squaring ensures that even if the coordinates result in a negative difference, the squared value is positive.
• Consistent Measurement: It maintains a consistent method by treating increases and decreases equally.

By summing these squared differences, we maintain accuracy before finally taking the square root to compute the overall distance.