Problem 87 - Algebra A Combined Function

Question

Write an ordered pair for each point described. Find the perimeter of the rectangle whose vertices are the points with coordinates \((-1,5),(3,5),(3,-4),\) and (-1,-4)

Step-by-Step Solution

Answer

The perimeter of the rectangle is 26 units.

Step 1: Understand Perimeter of a Rectangle

The perimeter of a rectangle is the total length around the rectangle, which can be calculated as \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width.

Step 2: Identify the Length and Width

For the vertices \((-1,5), (3,5), (3,-4), (-1,-4)\), the rectangle is aligned with the axes. The width (along the y-axis) can be calculated as the difference in y-coordinates \(5 - (-4) = 9\), and the length (along the x-axis) is the difference in x-coordinates \(3 - (-1) = 4\).

Step 3: Calculate the Perimeter

Use the formula for the perimeter of a rectangle: \( P = 2(l + w) \). Substitute the values found for length (\( l = 4 \)) and width (\( w = 9 \)) into the equation. Thus, \( P = 2(4 + 9) = 2 \times 13 = 26 \).

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around its edge. Imagine wrapping a string around the rectangle and how long that string would be.
To find the perimeter, you'll need to know the length of two sides: the length and the width. These are typically the longer and shorter sides of the rectangle, respectively. The formula to calculate the perimeter is:

• \( P = 2(l + w) \)

Where:

• \( l \) represents the length.
• \( w \) represents the width.

This formula comes from adding up all four sides of the rectangle. Since opposite sides of a rectangle are equal, you just need to double the sum of the length and width.
Take some time to practice this formula with different rectangles and remember, the perimeter measures the outer boundary.

Vertices of a Rectangle

Vertices are the points where the sides of a shape, like a rectangle, meet. Each vertex has a pair of coordinates, giving its position on a graph or coordinate plane.
A rectangle has four vertices, and knowing these helps us understand the shape's structure and size.
In a coordinate plane, the rectangle described has vertices at these points:

• \((-1, 5)\)
• \((3, 5)\)
• \((3, -4)\)
• \((-1, -4)\)

These ordered pairs tell us where each vertex is positioned, with the first number representing the x-coordinate and the second the y-coordinate.
Using these vertices, you can determine the rectangle's length and width by measuring the distance between appropriate pairs of points. This measurement is key to other calculations, such as finding the perimeter.

Ordered Pairs

Ordered pairs are like GPS coordinates for a point in the coordinate plane. They tell us exactly where to plot a point, using two numbers inside parentheses. The first number is the x-coordinate, which tells us how far left or right to go. The second is the y-coordinate, which tells us how far up or down to go.
For example, the ordered pair \((3, 5)\) means to go three units to the right from the origin and five units up.
In the context of the rectangle, each vertex is identified by an ordered pair:

• \((-1, 5)\)
• \((3, 5)\)
• \((3, -4)\)
• \((-1, -4)\)

These points define the corners of the rectangle. Understanding ordered pairs is crucial because they allow us to place and identify shapes on a coordinate plane easily.