Figures in the Coordinate Plane In this lesson, we will review the coordinate plane and plotting points in the plane. We will explore translations and reflections of 2dimensional figures. Finally, we will use the coordinate plane to find the dimensions and area of two dimensional figures.
The diagram pictured above is called the coordinate plane. The horizontal line labeled "x" is called the xaxis and the vertical line labeled "y" is called the yaxis. The point that is plotted has coordinates (1,3) since the point is 1 unit to the right and 3 units above the origin (the point where the xaxis and yaxis meet). A point is reflected across the xaxis if its xcoordinate stays the same and its ycoordinate is multiplied by 1. Graphically, we can think of a reflection across the xaxis as moving it to its mirror image where the mirror is the xaxis. Similarly, a point is reflected across the yaxis if its ycoordinate stays the same and its xcoordinate is multiplied by 1. Graphically, we can think of a reflection across the yaxis as moving it to its mirror image where the mirror is the yaxis. If we pick up a point and move it, we say that we are translating the point. We can also pick up a 2dimensional object and move it. We say that we are translating the object. Example 1 The front side of a pendant is shown below.
Solutions
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear. Exercise 1
What are the coordinates the image of B when the triangle ABC is reflected across the xaxis?
Example 2 A Tshirt print is to be made by taking the pattern below and including a second pattern that is 5 units to the right and 3 units below the first pattern. What will the new coordinates of point C be?
Solution Before translation, the point C has coordinates (4,1). Since we are moving the point 5 units to the right, the new xcoordinate is x = 4 + 5 = 1 and since we are moving the point 3 units down, the new ycoordinate is y = 1  3 = 2 So the new coordinates of the point C are (1,2). The sketch of the translated pattern is shown below. We can see that C does have coordinates (1,2).
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear. Exercise 2 Margarita wants to sketch two fish that are the same shape. She has already drawn one fish shown below. The second fish is to be drawn 2 units above and 4 units to the right of the first fish. What will be the coordinates of the new point A?
The CAHSEE often tests to see if you remember the names of the common polygons. Recall that a polygon is a closed shape bounded only by straight lines. Here are some of these definitions.
Example 3 The points (2,1), (4,1), (5,4), and (1,4) are the vertices of a polygon. What type of polygon is formed by these points? Solution First sketch the points on the xyplane. Then connect the points to form the polygon. The sketch is shown below.
Since the figure has 4 sides, it is not a triangle, pentagon, or a hexagon. The left and right sides are not parallel so it is not a parallelogram. The top and bottom sides have different lengths, so it is not a rhombus or a square. The angles are not right angles, so is it not a rectangle. Notice that the top and bottom are parallel, so this is a trapezoid. Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear. Exercise 3 The points (1,1), (3,1), (4,6), and (2,6) are the vertices of a polygon. What type of polygon is formed by these points? A. Triangle B. Trapezoid C. Parallelogram D. Pentagon
Another use of the coordinate plane is to find the lengths and area of the given geometric shape. Example 4 The graph of the square ABCD is shown below. What is the length of one of the sides of this square?
Solution Since ABCD is a square, all sides have the same length. We will find the distance from A to B. Counting, we see that B is exactly 3 units to the right of A. The length of the sides of the square are all 3 units. Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear. Exercise 4 The graph of the equilateral triangle ABC is shown below. What is the length of one of the sides of this equilateral triangle?
Example 5 The graph of the right triangle ABC is shown below.
Find the area of this triangle in square units.
Solution We use the formula for the area of a right triangle. Area = 1/2 bh where b is the length of the base of the triangle and h is the height of the triangle. The length of the base is the distance from A to B. We measure this as b = 5 units The height is the distance from A to C. We measure this as h = 3 units So that Area = (1/2)(5)(3) = 15/2 = 7.5 The area of the triangle is 7.5 square units. Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear. Exercise 5 The graph of the rectangle ABCD is shown below. What is the area in square units of this rectangle?
