Problem 10 - Precalculus

Question

Calculate the distance between the given points. \((-3,1) \text { and }(374,-335)\)

Step-by-Step Solution

Answer

The distance is 505 units.

Step 1: Understand the Distance Formula

The distance formula derived from the Pythagorean Theorem is used to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane. The formula is: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]

Step 2: Identify the Coordinates

Identify and label the coordinates of the given points. Let \((-3, 1)\) be the first point with \(x_1 = -3\) and \(y_1 = 1\). Let \((374, -335)\) be the second point with \(x_2 = 374\) and \(y_2 = -335\).

Step 3: Substitute into the Formula

Substitute the values of \(x_1, y_1, x_2,\) and \(y_2\) into the distance formula:\[d = \sqrt{(374 - (-3))^2 + (-335 - 1)^2}\]

Step 4: Calculate the Differences

Calculate the differences for both \((x_2 - x_1)\) and \((y_2 - y_1)\):\((x_2 - x_1) = 374 + 3 = 377\)\((y_2 - y_1) = -335 - 1 = -336\)

Step 5: Compute the Squares

Square the differences obtained:\((377)^2 = 142129\)\((-336)^2 = 112896\)

Step 6: Add the Squares

Add the squared results from the previous step:\[142129 + 112896 = 255025\]

Step 7: Take the Square Root

Take the square root of the sum to find the distance:\[d = \sqrt{255025} = 505\]

Step 8: Interpret the Result

The distance between the points \((-3, 1)\) and \((374, -335)\) is 505 units.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle of geometry that describes the relationship between the sides of a right triangle. Though originally used to solve problems involving triangles, it also provides a basis for the distance formula on a coordinate plane.
It states that in a right 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 can be formulated as:\[ c^2 = a^2 + b^2 \]where:

• \(c\) is the length of the hypotenuse
• \(a\) and \(b\) are the lengths of the other two sides

This theorem is the foundation for the distance formula because the distance between two points can be visualized as the hypotenuse of a triangle formed by the horizontal and vertical differences between the points.

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves using an ordered pair of numbers. The coordinate plane is formed by two perpendicular axes, usually referred to as the x-axis (horizontal) and y-axis (vertical).
Each point on the plane is described by a pair of numbers, usually written as \((x, y)\), where:

• \(x\) represents the horizontal position of the point
• \(y\) represents the vertical position of the point

When you are tasked with finding the distance between two points, these coordinates will play a crucial role as inputs in the distance formula.
Understanding how to navigate the coordinate plane is essential, as it helps you easily identify and locate the positions of various points, making calculations straightforward and less error-prone.

Distance Calculation

Distance calculation between two points on a coordinate plane involves finding the length of the line segment that directly connects the two points. This is done using the distance formula, which beautifully builds upon the principles of the Pythagorean Theorem.
Let's break it down into simple steps:

• First, identify the coordinates of the two points, \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the x-coordinates: \((x_2 - x_1)\).
• Subtract the y-coordinates: \((y_2 - y_1)\).
• Square each of these differences: \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\).
• Add the squared differences to form the sum: \[(x_2 - x_1)^2 + (y_2 - y_1)^2\]
• Finally, take the square root of this sum to get the distance \(d\): \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In our original exercise, this sequence of operations showed that the distance between the two points, \((-3,1)\) and \((374,-335)\), is 505 units. This method is reliable and allows for accurate computation of distance in any applicable scenario.