Problem 21 - Precalculus Mathematics for Calculus

Question

Plot the points \(A(1,0), B(5,0), C(4,3),\) and \(D(2,3)\) on a coordinate plane. Draw the segments \(A B, B C, C D,\) and \(D A\) What kind of quadrilateral is \(A B C D,\) and what is its area?

Step-by-Step Solution

Answer

Quadrilateral \(ABCD\) is a trapezoid with area 9 square units.

Step 1: Plot Points on Coordinate Plane

To plot points on a coordinate plane, start by marking each point using its respective coordinates. For point \(A(1,0)\), move 1 unit right on the x-axis. For point \(B(5,0)\), move 5 units right on the x-axis. Point \(C(4,3)\) lies 4 units right and 3 units up. Finally, plot point \(D(2,3)\) by moving 2 units right and 3 units up.

Step 2: Draw Line Segments

Connect the points with straight lines to form the quadrilateral. Draw segment \(AB\) connecting points \(A\) and \(B\), segment \(BC\) connecting \(B\) and \(C\), segment \(CD\) connecting \(C\) and \(D\), and segment \(DA\) connecting \(D\) and \(A\).

Step 3: Identify Quadrilateral Type

Examine the lengths of the sides and the angles. With both points \(C\) and \(D\) having the same y-coordinate and both points \(A\) and \(B\) having the same y-coordinate, we see a trapezoid because one pair of opposite sides (\(AB\) and \(CD\)) are parallel.

Step 4: Calculate the Area of the Trapezoid

Use the formula for the area of a trapezoid: \(Area = \frac{1}{2} \times (b_1 + b_2) \times h\), where \(b_1\) and \(b_2\) are the lengths of the parallel sides, and \(h\) is the height. Here, \(b_1 = AB = 4\), \(b_2 = CD = 2\), and the height \(h\) is the vertical distance between the parallel sides, which is 3 units. So, \(Area = \frac{1}{2} \times (4 + 2) \times 3 = 9\).

Key Concepts

Understanding Quadrilateral

A quadrilateral is a four-sided polygon, which can vary in shape and form. Each quadrilateral has four vertices (corners) and four edges (sides). Common examples include squares, rectangles, and trapezoids.
For a figure to be classified as a specific type of quadrilateral, it needs to meet certain criteria based on the arrangement of its sides and angles.

• **Square and Rectangle:** Opposite sides are equal in length and all angles are 90 degrees.
• **Trapezoid:** At least one pair of opposite sides are parallel.
• **Rhombus:** All sides are equal, but angles are not required to be 90 degrees.

The quadrilateral in this exercise, labeled as \(ABCD\), is identified as a trapezoid. This is because the segments \(AB\) and \(CD\) are parallel.

Techniques for Plotting Points

Plotting points on a coordinate plane is an essential skill in geometry that allows you to visualize shapes like quadrilaterals.
Each point is represented by an ordered pair \((x, y)\) known as coordinates.

• The **x-coordinate** tells you how far to move along the x-axis (horizontally).
• The **y-coordinate** indicates your movement along the y-axis (vertically).

For instance, to plot the point \((1, 0)\), you move 1 unit to the right on the x-axis while staying on the x-axis since the y-coordinate is 0. Consistently following this method for all given points helps you to accurately map the vertices of a quadrilateral on the coordinate plane.

Approaching Area Calculation

Finding the area of a quadrilateral, like a trapezoid, can be done using geometry formulas tailored to specific shapes. In this case, we use the formula for the area of a trapezoid:
To recap, the area of a trapezoid is calculated as follows:\[Area = \frac{1}{2} \times (b_1 + b_2) \times h\]Here,

• \(b_1\) and \(b_2\) are the lengths of the parallel sides.
• \(h\) is the height, or the perpendicular distance between these sides.

Apply the formula by substituting \(b_1 = 4\), \(b_2 = 2\), and \(h = 3\).
The calculation yields an area of 9 square units, showing the space the quadrilateral occupies on the plane.

Exploring the Coordinate Plane

The coordinate plane is formed by two number lines intersecting at right angles, creating a grid. The vertical line is the y-axis, and the horizontal line is the x-axis. Together, they allow you to plot points, lines, and shapes.

By using the coordinate plane, you can easily:

• Identify and plot various geometric shapes and figures.
• Establish relationships between points and recognize patterns.
• Measure distances and angles accurately.

Plotting the quadrilateral within the coordinate plane offers a clear representation of its structure, enabling accurate analysis and manipulation.