The task in the image requires solving the following steps:

1. Draw a coordinate plane.
2. Mark points:
   A(5;4), B(3;-2), C(-5;7), and D(-8;-1).
3. Find the projection of movement on the axes and calculate the distance between the points.
4. Input values with proper signs and round to hundredths.

Solution Approach:

1. Coordinate Plane and Points

• Draw a two-dimensional Cartesian plane (x and y axes).
• Mark the given points:
  • $A(5, 4)$
  • $B(3, -2)$
  • $C(-5, 7)$
  • $D(-8, -1)$

2. Projection of Movement

To calculate the movement (displacement) between points, find the projection on both the x and y axes for each pair of points. The general formula for displacement $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Between A and B: $x_{\text{proj}} = 3 - 5 = -2, \quad y_{\text{proj}} = -2 - 4 = -6$ Displacement between A and B: $d_{AB} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32$

• Between B and C: $x_{\text{proj}} = -5 - 3 = -8, \quad y_{\text{proj}} = 7 - (-2) = 9$ Displacement between B and C: $d_{BC} = \sqrt{(-8)^2 + 9^2} = \sqrt{64 + 81} = \sqrt{145} \approx 12.04$

• Between C and D: $x_{\text{proj}} = -8 - (-5) = -3, \quad y_{\text{proj}} = -1 - 7 = -8$ Displacement between C and D: $d_{CD} = \sqrt{(-3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54$

3. Final Results:

• The projections for each axis for every pair of points are calculated.
• The total displacements have been rounded to hundredths as required.

Would you like any more details or additional clarification on the steps?

Follow-up Questions:

1. How do you calculate the displacement for other points on the coordinate plane?
2. What are the formulas used to find the projections of movement along axes?
3. How would the result change if one of the points were in 3D space?
4. What are the properties of the Cartesian plane used in this calculation?
5. Can you calculate displacement using a different method or visual tool?

Tip:

When plotting points, always check if the coordinates are positive or negative, to place them correctly on the x and y axes.