Problem 19 - Precalculus Mathematics for Calculus

Question

Draw the rectangle with vertices \(A(1,3), B(5,3), C(1,-3)\) and \(D(5,-3)\) on a coordinate plane. Find the area of the rectangle.

Step-by-Step Solution

Answer

The area of the rectangle is 24 square units.

Step 1: Plot the Vertices

First, place the given vertices on a coordinate plane. The vertices are points where the sides of the rectangle intersect. - Point A is at (1,3). - Point B is at (5,3). - Point C is at (1,-3). - Point D is at (5,-3).

Step 2: Draw the Rectangle

Connect the plotted vertices to form the rectangle. Connect A to B, B to D, D to C, and finally C back to A. This will outline the rectangle ABCD on the coordinate plane.

Step 3: Calculate the Length

Find the length of the rectangle using points A and B. Since A and B have the same y-coordinate, measure the horizontal distance between them: Length (AB) = |x2 - x1| = |5 - 1| = 4.

Step 4: Calculate the Width

Find the width of the rectangle using points A and C. Since A and C have the same x-coordinate, measure the vertical distance between them: Width (AC) = |y2 - y1| = |3 - (-3)| = 6.

Step 5: Calculate the Area

The area of a rectangle is given by Area = length × width. Substitute the values we have found: Area = 4 × 6 = 24.

Key Concepts

Coordinate Plane

Imagine a large blank sheet divided into squares, like a giant graph paper that extends in every direction. This is the coordinate plane. It's a two-dimensional surface where each point is represented by a pair of numbers. The first number is the position on the horizontal line (x-axis), and the second number is the position on the vertical line (y-axis). Each pair \(x, y\) corresponds to a unique point.
Some key things to remember about the coordinate plane:

• The horizontal line is called the x-axis, and the vertical line is called the y-axis.
• Where they intersect is called the origin, with coordinates \(0,0\).
• Points are plotted as \(x, y\), moving right or left along the x-axis and up or down along the y-axis.

Rectangle

A rectangle is a four-sided polygon, also known as a quadrilateral. It has some specific properties that make it unique:

• Each of its four sides is straight.
• Opposite sides are equal in length.
• All interior angles are right angles, each measuring 90 degrees.

These properties make a rectangle not only visually distinctive but also easy to analyze geometrically, widely used in both daily applications and mathematical contexts.

Vertices

The term 'vertices' refers to the corners or points where two or more lines meet. In the context of a rectangle, the vertices are the four points where the rectangle's sides intersect. Each vertex is defined by coordinates on a coordinate plane.
For our rectangle, the vertices are labeled as:

• Vertex A at \(1,3\).
• Vertex B at \(5,3\).
• Vertex C at \(1,-3\).
• Vertex D at \(5,-3\).

These points are crucial because, once connected, they form the boundary of the rectangle.

Area Calculation

The area of a rectangle is a measure of the space it occupies within its perimeter. When calculating the area, we use the formula: \[ \text{Area} = \text{Length} \times \text{Width} \]For our rectangle:

• The length is the distance between vertices A and B, which is the horizontal span. The calculation is straightforward: \|5 - 1| = 4\.
• The width is the distance between vertices A and C, which measures the vertical span. Calculate by \|3 - (-3)| = 6\.

Multiply the length by the width to find the area: \4 \times 6 = 24\.
This formula is essential as it applies to any rectangle, making it a handy tool for everyday and mathematical use.