Coordinate System, Distance Formula, and Geometric Shapes Problems

This assignment delves into the concepts of the Cartesian coordinate system and the distance formula, providing solutions to a range of geometric problems. The assignment begins with exercises involving plotting points, determining if they form triangles, and identifying the coordinates of shapes. It progresses to problems calculating distances between points and finding the radius of circles. Furthermore, it explores geometric concepts such as complementary angles, deductive reasoning, and properties of triangles. The assignment includes problems on calculating areas and perimeters of various shapes, including octagons and hexagons, and solving for unknown angles and lengths in geometric figures. It also covers concepts related to cylinders, cones, and contrapositive statements, offering a comprehensive review of fundamental geometric principles. The solutions are presented in a step-by-step manner, including diagrams, calculations, and explanations, making it an excellent resource for students seeking to understand and master these concepts. The document showcases different shapes, their properties and also touches upon trigonometry to solve for lengths and angles.

The Cartesian coordinate system and the Distance Formula
Solve the problems below. Be sure to show all of your work. We suggest saving your work
in a word processor.
Exercises
Draw a Cartesian Coordinate system for yourself, and then determine whether these sets of
points form triangles
1. Set 1: (0, 0), (1, 1), (0, 1)
Set 2: (-1, 0), (1, 1), (3, 2)
Set 3: (0, 0), (5, 0), (3, 5)
Solution:
Set 1: (0, 0), (1, 1), (0, 1)

From the above figure, it is observed that the points (0, 0), (1, 1), (0, 1) form triangle.
Set 2: (-1, 0), (1, 1), (3, 2)

From the above figure, it is observed that the points (-1, 0), (1, 1), (3, 2) form triangle.
From the above figure, it is observed that the points (-1, 0), (1, 1), (3, 2) form triangle. They are
on the same line and form straight line.
Set 3: (0, 0), (5, 0), (3, 5)

2. I have placed a square that is 2 units long on each side so that its sides are parallel to the x- or
y-axis, but its center is right on the origin. What are the coordinates of the corners?
Solution:

From the above graph, the corner points of the square are (2, 2), (-2, 2), (-2, -2) and (2, -2).
3. What shape do these four points make? (Draw from one to the next in the order given, and
then draw from the last back to the first).
(0,-1), (1, 1), (0, 3), (-1, 1)
Solution: The graph from the given points is
And the above shape is a rhombus.

4. Now do the same with this list, and describe the shape.
(0,-1), (0, 3), (1, 1), (-1, 1)
Solution: The graph from the given points is
And the shape is shown above.
What shape is created by the following points? (Do not connect these in order; just connect
around the outside of the shape.)
5. (-1, 0), (2, 0), (2, 4), and (-1, 4)
Solution:

The above figure is a rectangle.
6. (-2, 1), (-6, 1), (-7, 6), and (-3, 6)
Solution:

7. (1, 1), (1, -3), (9, 1), and (5, -3)
Solution:

8. (8, 5), (2, -3), (5, -3), and (5, 5)
Solution:

9. (-5, 1), (2, 1), (2, -4), and (-13, -4)
Solution:

10. (-7, 9), (-3, 13), and (-11, 13)
Solution:

Find the distance between the two points:
11. (-20, -4) and (-7, -6)
12. (1, 1) and (-4, 1)
13. (-3, 22) and (-14, 35)
14. (9, -0) and (-3, -8)
15. (1, 2) and (5, 2)
Solution: We know that the distance between two points is
11. The distance between (-20, -4) and (-7, -6) is