Problem 8 - Prealgebra

Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. $$V(8,-5), W(-3,-5)$$

Step-by-Step Solution

Answer

The distance between points V and W is 11 units.

Step 1: Understand the Distance Formula

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) in the coordinate plane can be found using the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] This formula derives from the Pythagorean theorem.

Step 2: Identify the Coordinates

Identify the coordinates of the given points. For point \(V\), the coordinates are \(x_1 = 8, y_1 = -5\). For point \(W\), the coordinates are \(x_2 = -3, y_2 = -5\).

Step 3: Substitute into the Distance Formula

Substitute the coordinates of \(V\) and \(W\) into the distance formula: \[d = \sqrt{((-3) - 8)^2 + ((-5) - (-5))^2}\]

Step 4: Simplify the Expressions

Calculate the expressions inside the square root: \((-3) - 8 = -11\) and \(-5 - (-5) = 0\). So the expression becomes: \[d = \sqrt{(-11)^2 + (0)^2}\]

Step 5: Solve the Squared Terms

Calculate the squares: \((-11)^2 = 121\) and \(0^2 = 0\). Substituting back, we get: \[d = \sqrt{121 + 0}\]

Step 6: Compute the Square Root

Find the square root of 121: \[\sqrt{121} = 11\] This is the distance between the points \(V(8,-5)\) and \(W(-3,-5)\).

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. 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. Mathematically, this is expressed as:\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides. This theorem is not just limited to triangles; it's the backbone of the distance formula we often use in the coordinate plane. By treating the horizontal and vertical distances between two points as the sides of a right triangle, we can apply the Pythagorean theorem to find the distance between them. This makes it extremely useful in various applications such as navigation, surveying, and even in computer graphics.
Understanding this theorem gives us a deeper insight into how distances operate in both 2D and 3D spaces, making it a versatile tool for solving practical and theoretical problems. Remember, when you calculate distance with the Pythagorean theorem, you essentially "uncover" the invisible hypotenuse in the grid of the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It is divided into four quadrants by two perpendicular lines called axes. The horizontal axis is known as the \( x \)-axis, and the vertical axis is the \( y \)-axis. The intersection point of these two axes is called the origin, denoted as \((0,0)\).
Every point in the coordinate plane is identified with an ordered pair of numbers \((x, y)\), which are called coordinates. The first number, \(x\), is the horizontal position, and the second number, \(y\), is the vertical position of the point.

• Points in the first quadrant have positive \(x\) and \(y\) coordinates.
• Points in the second quadrant have negative \(x\) coordinates and positive \(y\) coordinates.
• In the third quadrant, both \(x\) and \(y\) coordinates are negative.
• In the fourth quadrant, the \(x\) coordinates are positive, but the \(y\) coordinates are negative.

Using the coordinate plane, we can visually see the relationship between points and easily calculate distances or slopes. It's a powerful tool for visualizing algebraic equations and understanding geometry. It helps us form connections between visual and numerical data, making math more understandable and applicable to real-world situations.

Square Root Calculation

Square root calculation is a mathematical operation that undoes squaring a number. If you have a number like 25 and you want to know what number makes 25 when multiplied by itself, you are looking for the square root, which in this case is 5. In symbols, the square root is denoted by the radical sign \( \sqrt{} \).
For example:

• \( \sqrt{25} = 5 \)
• \( \sqrt{36} = 6 \)

The square root is central in the calculation of distances, as seen in the distance formula where you solve for the root of squared differences. To compute a square root by hand:

• Estimate the root by finding the nearest perfect square numbers.
• Refine your estimate by trial and error to get closer to an exact root.

When dealing with non-perfect squares, sometimes it is necessary to use a calculator due to their more complex solutions. Understanding square roots is crucial not only in math problems but also in physics, engineering, and even in financial modeling, where determining the root of an equation helps unravel the answers to more profound questions.