Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Plot the points \(A(-3,1), B(3,1), C(-2,-3), D(0,3)\), and \(E(2,-3)\) on a coordinate plane. Draw the line segments \(A B\), \(B C, C D, D E\), and \(E A\).

Step-by-Step Solution

Answer

Plot the points, then connect them in order with line segments.

Step 1: Plotting Point A

Locate point A by moving 3 units to the left along the x-axis (since the x-coordinate is -3), and then 1 unit up along the y-axis (since the y-coordinate is 1). Place a point at (-3, 1).

Step 2: Plotting Point B

Locate point B by moving 3 units to the right along the x-axis, and staying at the same level on the y-axis since the y-coordinate is 1. Place a point at (3, 1).

Step 3: Plotting Point C

Locate point C by moving 2 units to the left along the x-axis and 3 units down along the y-axis since the coordinates are (-2, -3). Place a point at (-2, -3).

Step 4: Plotting Point D

Locate point D by not moving along the x-axis (since the x-coordinate is 0) and moving 3 units up along the y-axis. Place the point at (0, 3).

Step 5: Plotting Point E

Locate point E by moving 2 units to the right along the x-axis and 3 units down along the y-axis. Place the point at (2, -3).

Step 6: Drawing Line Segment AB

Draw a straight line from point A to point B using a ruler or straightedge.

Step 7: Drawing Line Segment BC

Draw a straight line from point B to point C using a ruler or straightedge.

Step 8: Drawing Line Segment CD

Draw a straight line from point C to point D using a ruler or straightedge.

Step 9: Drawing Line Segment DE

Draw a straight line from point D to point E using a ruler or straightedge.

Step 10: Drawing Line Segment EA

Draw a straight line from point E back to point A using a ruler or straightedge to complete the shape.

Key Concepts

Plotting Points

Plotting points on a coordinate plane might seem complex at first, but it's as simple as playing a game of connect-the-dots. When plotting a point such as (-3,1), the first number, -3, tells you to move on the x-axis, which is the horizontal line. This movement is to the left because it's negative. The second number, 1, tells you to move on the y-axis, which is the vertical line, and since it’s positive, you move up. It's like giving directions using a map! For each point, follow these steps:

• Check the x-coordinate to decide how far to move horizontally.
• Check the y-coordinate to decide how far to move vertically.
• Combine these movements and place your point where they meet.

By practicing these steps, plotting points becomes intuitive.

Line Segments

Line segments are essentially straight paths connecting two points. When you're given points such as A and B, drawing a line segment means joining these two dots with a straight line. Imagine connecting them with a piece of string pulled tight to ensure straightness. Here's a simple way to approach drawing line segments:

• Identify the two points you need to connect.
• Use a straightedge like a ruler to align with both points.
• Draw a straight line starting from the first point to the second.

This technique ensures accuracy and neatness in your drawing, creating perfect line segments every time.

Coordinate Plane

The coordinate plane is like a giant blank canvas divided into four sections by two crossing lines. These lines are called axes: the x-axis (horizontal) and the y-axis (vertical). Where they intersect is known as the origin, labeled as (0,0). This system helps you precisely locate points in a plane just like finding coordinates on a map. Here are the key components:

• Axis: The reference lines—x-axis and y-axis.
• Quadrants: The four sections created by the axes intersection. Each quadrant signifies a specific combination of positive/negative numbers on x and y.
• Origin: The center point of the plane (0,0).

Understanding this layout makes using coordinate planes much easier, as it provides a structured method to display and connect points.

Drawing Lines

Drawing lines on a coordinate plane goes hand-in-hand with plotting points and using line segments. After plotting your points, the next step is to connect them in a specific order to create shapes or paths, as needed. Even when not connecting them in pairs like segments, following their order properly is crucial for accuracy:

• Layout your points based on their coordinates on the plane.
• Utilize a straightedge to connect the points, ensuring precision.
• Double-check each line begins and ends at the correct points in the given sequence.

By consistently applying these steps, you can easily create accurate lines that represent paths, shapes, or even intricate designs on the coordinate plane.