Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. \(G(-6,8), H(-6,-4)\)

Short answer

The distance is 12 units.

Step-by-step solution

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: \( ext{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). This formula is derived from the Pythagorean theorem.

Substitute Coordinates into Formula

Identify the coordinates of points \(G\) and \(H\). Here, \(G = (-6, 8)\) and \(H = (-6, -4)\). Substitute these values into the distance formula: \( ext{Distance} = \sqrt{((-6) - (-6))^2 + ((-4) - 8)^2}\).

Simplify Expressions Inside the Square Root

Calculate the differences for \(x\) and \(y\) coordinates: - For x-coordinates: \((-6) - (-6) = 0\).- For y-coordinates: \((-4) - 8 = -12\).Therefore, the expression becomes \( ext{Distance} = \sqrt{0^2 + (-12)^2}\).

Compute the Squares

Now calculate the squares: - \(0^2 = 0\).- \((-12)^2 = 144\).This results in \( ext{Distance} = \sqrt{0 + 144}\ = \sqrt{144}\).

Calculate the Square Root

Find the square root of 144: \( ext{Distance} = \sqrt{144} = 12\). Thus, the distance between points \(G\) and \(H\) is 12 units.

Key concepts

Coordinate Plane

The coordinate plane is a fundamental element in geometry. It consists of two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is known as the origin, marked as point (0,0). This system allows us to represent any point in a two-dimensional space with an ordered pair of numbers (x, y), known as coordinates.

In the given exercise, the points are presented as coordinates - specifically, point \( G \) with coordinates \((-6, 8)\) and point \( H \) with coordinates \((-6, -4)\). The x-coordinate indicates horizontal movement from the origin, while the y-coordinate indicates vertical movement. This setup makes it easier to analyze the positions of points and calculate distances between them.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry dealing with right triangles. It states that for 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 is expressed with the formula: \[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.

The distance formula, used to find the distance between two points on a coordinate plane, is directly derived from the Pythagorean theorem. By interpreting the difference in x-coordinates and y-coordinates as the legs of a right triangle, the distance between the points becomes the hypotenuse. Thus, in the context of our points \( G \) and \( H \), we use this theorem to calculate the distance effectively.

Distance Calculation

Calculating the distance between two points on a coordinate plane utilizes the distance formula. This formula is:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Let's see this in action using our points \( G(-6, 8) \) and \( H(-6, -4) \):

• First, find the difference between the x-coordinates: \((-6) - (-6) = 0\).
• Then, find the difference between the y-coordinates: \((-4) - 8 = -12\).

Next, plug these differences into the formula: \[ \text{Distance} = \sqrt{0^2 + (-12)^2} \]After squaring these expressions, you compute:

• \(0^2 = 0\)
• \((-12)^2 = 144\).

Finally, the distance equals:\[ \text{Distance} = \sqrt{144} = 12 \]Therefore, the points \( G \) and \( H \) are 12 units apart.

Geometry Basics

Basics of geometry such as points, lines, and planes form the foundation for understanding more complex concepts. Here's a refresher:

• Points are locations in space, usually defined by coordinates.
• Lines are straight paths that extend infinitely in both directions. They connect points.
• Planes are flat surfaces that extend infinitely in all directions; they contain lines and points.

In this exercise, the given points and the distance calculated between them rely on these basics. Each point is like a marker on this vast, conceptual plane, and understanding their relationships (like distance) is at the heart of learning geometry. The coordinate plane we discussed earlier is simply a plane with specific numerical guides to pinpoint the location of each point.