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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 xx and yy coordinates to determine their location.

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

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

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

Identifying the Opposite of a Number

Problem 2

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

  • The opposite of −3-3 is 33. To find this on the number line, you look for the point that is equidistant from 00 but in the opposite direction. In this case, it’s 33, so the correct point is C\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∘30^\circ (360° divided by 12 hours). The angle between 8 and 12 is 4×30∘=120∘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∘−120∘=240∘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∘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 (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on a coordinate plane is given by:

Distance=(x2−x1)2+(y2−y1)2\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

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

  1. Find the difference in the xx-coordinates: −5−(−6)=1-5 - (-6) = 1.

  2. Find the difference in the yy-coordinates: −4−4=−8-4 - 4 = -8.

  3. Plug these differences into the distance formula:

    Distance=(1)2+(−8)2=1+64=65\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.

 

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