Problem 11 - Intermediate Algebra

Question

Fill in the blanks. The formula to find the distance between points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) \text { is } d=\sqrt{\square+\square}\)

Step-by-Step Solution

Answer

The formula is: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).

Step 1: Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem and is used to find the distance between two points on a coordinate plane.

Step 2: Analyzing the Formula Structure

The distance formula is structured as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula calculates the distance 'd' by taking the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates.

Step 3: Filling in the Blanks

Based on the formula structure:- The first blank should contain \((x_2 - x_1)^2\).- The second blank should contain \((y_2 - y_1)^2\).

Step 4: Complete the Formula

The complete formula for the distance 'd' between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates to right-angled triangles. 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 relationship is written as: \[ c^2 = a^2 + b^2 \] where \(c\) is the hypotenuse, while \(a\) and \(b\) are the triangle's other two sides.
In the context of the distance formula, the coordinate plane acts as a series of right triangles where the difference in x-values and y-values represents \((a)\) and \((b)\). The calculated distance acts as the hypotenuse \((c)\). Therefore, when plotting two points on a plane, you can imagine a right triangle forming between them. The base and height of this triangle are parallel to the x-axis and y-axis, making it easy to apply the Pythagorean Theorem.

Coordinate Plane

A coordinate plane is a two-dimensional plane formed by two perpendicular axes, usually labeled as the x-axis and the y-axis. These axes intersect at a point called the origin, denoted by \((0,0)\).
The coordinate plane allows us to plot points and understand their spatial relationships. Each point on this plane is represented by a pair of numerical coordinates: \((x, y)\).

• The x-coordinate shows the point's horizontal position relative to the origin.
• The y-coordinate indicates the vertical positioning.

This system of describing positions helps us calculate distances between points straightforwardly by applying the distance formula derived from the Pythagorean Theorem.

Distance Calculation

Distance Calculation, especially on a coordinate plane, means finding how far two points are apart. The distance formula helps us do this simply and efficiently by connecting concepts from geometry directly with algebra.
The distance formula, derived from the Pythagorean Theorem, uses the differences between x-coordinates and y-coordinates to construct a right triangle. To find the distance \(d\), you follow this formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here's how it works:

• Subtract the x-values of the two points to get the horizontal difference, \((x_2 - x_1)\).
• Subtract the y-values to get the vertical difference, \((y_2 - y_1)\).
• Square both differences: \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\).
• Add these squares together.
• The final step is to take the square root of this sum to get the distance \(d\).

By following these steps, you find the straight line distance between any two points on a plane, which is an essential part of geometry and many application areas, such as navigation and mapping.