Problem 35 - Algebra and Trigonometry

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 on the coordinate plane

Start by locating and plotting each of the vertices \(A(1,3)\), \(B(5,3)\), \(C(1,-3)\), and \(D(5,-3)\) on the coordinate plane. These points will help form the rectangles by marking out its corners.

Step 2: Connect the vertices to form the rectangle

Draw straight lines to connect the points in the following order: \(A\) to \(B\), \(B\) to \(D\), \(D\) to \(C\), and \(C\) to \(A\). This completes the rectangle on the coordinate plane.

Step 3: Determine the length of the sides of the rectangle

Use the coordinates to find the length of the sides. The horizontal sides are from \(A(1,3)\) to \(B(5,3)\) and \(C(1,-3)\) to \(D(5,-3)\). Calculate the horizontal length: \( |5 - 1| = 4 \). The vertical sides are from \(A(1,3)\) to \(C(1,-3)\) and \(B(5,3)\) to \(D(5,-3)\). Calculate the vertical length: \( |3 - (-3)| = 6 \).

Step 4: Calculate the area of the rectangle

To find the area of the rectangle, multiply the length of the horizontal side by the length of the vertical side. This calculation is: \( Area = 4 \times 6 = 24 \). Therefore, the area of the rectangle is 24 square units.

Key Concepts

area calculation

Calculating the area of a rectangle on a coordinate plane involves applying a simple formula. The area of a rectangle is determined by multiplying the length of its base by its height. On a coordinate plane, these dimensions correspond to the differences in the x-coordinates (for the base) and y-coordinates (for the height).

For the given rectangle with vertices at \(A(1,3), B(5,3), C(1,-3),\) and \(D(5,-3)\):

• The base, which lies between points \(A\) and \(B\), can be measured by finding the difference between their x-coordinates: \(|5 - 1| = 4\).
• The height, between points \(A\) and \(C\), can be found by the difference in their y-coordinates: \(|3 - (-3)| = 6\).
• Thus, the area is calculated as \(4 \times 6 = 24\) square units.

This simple process illustrates why understanding how to calculate differences between coordinates is crucial in coordinate geometry.

vertices plotting

Plotting vertices on a coordinate plane is the first crucial step in understanding the shape and properties of geometric figures. A coordinate plane consists of two axes: the horizontal x-axis and the vertical y-axis, which intersect at the origin (0,0).

For the rectangle with vertices \(A(1,3), B(5,3), C(1,-3),\) and \(D(5,-3)\):

• Each point has a coordinate pair, where the first number is the x-coordinate and the second is the y-coordinate.
• To plot \(A(1,3)\), move 1 unit along the x-axis and 3 units up the y-axis.
• Continue similarly for \(B(5,3)\), \(C(1,-3)\), and \(D(5,-3)\).

Accurately plotting these points is necessary to ensure the figure is drawn correctly, enabling the subsequent steps in problem-solving, such as determining side lengths and calculating area.

rectangle properties

Rectangles have distinct properties that are important in geometry and are easily observable when using coordinate geometry. A rectangle is a quadrilateral with four right angles. Its opposite sides are equal and parallel.

When dealing with rectangles on a coordinate plane:

• The coordinates help determine the orientation and side lengths of the rectangle.
• In our example, vertices \(A(1,3), B(5,3), C(1,-3),\) and \(D(5,-3)\) form a rectangle due to equal and parallel opposing sides: \((AB = CD)\) and \((AC = BD)\).
• A key property is that the diagonals of a rectangle are equal in length; this can be verified by calculating the distance between opposite vertices, such as \(AD\) and \(BC\).

Recognizing these properties is fundamental in identifying and working with rectangles accurately on a coordinate plane, highlighting their symmetry and simplicity.