Identifying and Plotting Points on the Coordinate Plane

Identifying and Plotting Points on the Coordinate Plane

Problem 1.1

When identifying points on the coordinate plane, you use the $x$ and $y$ coordinates to determine their location.

• Point A is described as 3 units to the left of the origin (negative $x$ direction) and 1 unit below the origin (negative $y$ direction). This places Point A at $(-3, -1)$.

• Point B is described as 3 units to the right of the origin (positive $x$ direction) and 1 unit below the origin (negative $y$ direction). This places Point B at $(3, -1)$.

• Point C is 1 unit to the left of the origin (negative $x$ direction) and 3 units above the origin (positive $y$ direction). This places Point C at $(-1, 3)$.

Identifying the Opposite of a Number

Problem 2

On the number line, the opposite of a number $a$ is $-a$. This means:

• The opposite of $-3$ is $3$. To find this on the number line, you look for the point that is equidistant from $0$ but in the opposite direction. In this case, it’s $3$, so the correct point is $\blueE{C}$.

Estimating Angle Measurement

Problem 3.1

To estimate angles, especially those similar to clock hands:

• The angle between the 8 and the 12 on a clock face can be calculated by considering that each hour mark represents $30^\circ$ (360° divided by 12 hours). The angle between 8 and 12 is $4 \times 30^\circ = 120^\circ$.

• However, the problem is asking for the longer rotation, which is the angle you get by going the other way around the clock. This would be $360^\circ - 120^\circ = 240^\circ$. To get the closest provided option, we estimate the longer rotation between the clock hands:

  • If we measure the angle in the other direction, it's approximately $310^\circ$ (360° minus the smaller angle), which matches the closest provided choice.

Calculating Distance with the Pythagorean Theorem

Problem 4

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a coordinate plane is given by:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

For the points $(-6, 4)$ and $(-5, -4)$:

1. Find the difference in the $x$-coordinates: $-5 - (-6) = 1$.

2. Find the difference in the $y$-coordinates: $-4 - 4 = -8$.

3. Plug these differences into the distance formula:

   $$\text{Distance} = \sqrt{(1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}$$

This result tells us how far apart the two points are.

How These Concepts Apply to Transformations

• Plotting Points: Understanding the positions of points helps in visualizing and performing transformations (translations, rotations, etc.) on a coordinate plane.

• Number Opposites: Knowing how to find the opposite of a number is crucial for reflecting points across axes and understanding rotations.

• Estimating Angles: Estimating angles helps in performing and understanding rotations. For example, knowing how much to rotate a figure around a point.

• Distance Calculation: Distance calculations are used in transformations like dilation (scaling) to understand how much a figure's size changes in relation to a center of dilation.