L4 Predicting the Coordinates Examples
Completion requirements
Unit A: Geometry
Chapter 2: Transformations
Answer the following questions using point P in the diagram.
a.
State the coordinates of point P.
b.
State the coordinates of the image, P', when point P is rotated 90° counterclockwise about the origin.
c.
State the coordinates of the image, P', when point P(–5, –3) is rotated 180° counterclockwise about the origin.
d.
State the coordinates of the image, P', when point P(–5, –3) is rotated 270° counterclockwise about the origin.
e.
State the coordinate of the image, P', when point P(–5, –3) is rotated 360° counterclockwise about the origin.
a.
P(–5, –3)
b.
To find the coordinates of the image, the paper can be rotated. The original point, P(–5, –3), can also be used. To find the coordinates of the image, first switch the x and y values. The image is in quadrant
4 (bottom right), so
x is positive and y is negative. The coordinates of the image, P', are (3, –5), as shown on the following graph.
c.
To find the coordinates, the paper can be rotated. Another option is to use the original point, P(–5, –3). When a 180° rotation occurs, x and y values of the image remain the same as the original point; however,
the signs of the
x and y values of the image are the
opposite to the x and y values of the original point. The image is in quadrant 1 (top right), so the x value is positive and the y value is positive. The coordinates of the image,
P', are (5, 3), as shown on the following graph.
d.
To find the coordinates of the image, the paper can be rotated. Another option is to use the original point, P(–5, –3). Switch the x and y values of the original point. The image is in quadrant 2 (top left),
so
x is negative and y is positive. The coordinates of the image, P', are (–3, 5), as shown on the following graph.
e.
When a point is rotated 360°, the image is in the same location as the original. The coordinates of the image, P', are (–5, –3).
Rotate quadrilateral GHIJ 180° counterclockwise about the origin.
a.
State the coordinate of the image, quadrilateral G'H'I'J'.
b.
Draw and label the image, G'H'I'J'.
a.
Step 1: Write the coordinates of the original quadrilateral: G(2, –4), H(6, –2), I(11, –6), J(8, –6).
Step 2: Determine which quadrant the image will appear in. After the 180° rotation counterclockwise (to the left), the image will be in quadrant 2. In quadrant 2, the x values are negative and the y values are positive.
Step 3: When a 180° rotation occurs, the x and y values of the image remain the same as the original point; however, the signs of the x and y values of the image are opposite to the x and y values of the original point. The coordinates of the image are G'(–2, 4), H'(–6, 2), I'(–11, 6), J'(–8, 6).
Step 2: Determine which quadrant the image will appear in. After the 180° rotation counterclockwise (to the left), the image will be in quadrant 2. In quadrant 2, the x values are negative and the y values are positive.
Step 3: When a 180° rotation occurs, the x and y values of the image remain the same as the original point; however, the signs of the x and y values of the image are opposite to the x and y values of the original point. The coordinates of the image are G'(–2, 4), H'(–6, 2), I'(–11, 6), J'(–8, 6).
b.
Rotate quadrilateral WXYZ 270° clockwise about the origin.
Note: The identical image will be produced if quadrilateral WXYZ is rotated 90° counterclockwise.
a.
State the coordinate of the image, quadrilateral W'X'Y'Z'.
b.
Draw and label the image, W'X'Y'Z'.
a.
Step 1: Write the coordinates of the original quadrilateral:
W(–9, –4), X(–5, –3), Y(–5, –6), Z(–9, –8)
Step 2: Determine which quadrant the image will appear in. After the 270° rotation clockwise (to the right), the image will be in quadrant 4. In quadrant 4, the x values are positive and the y values are negative.
Step 3: Switch the x and y values in each coordinate. Include the correct signs based on the location of the image. The coordinates for the image are W'(4, –9), X'(3, –5), Y'(6, –5), Z'(8, –9).
W(–9, –4), X(–5, –3), Y(–5, –6), Z(–9, –8)
Step 2: Determine which quadrant the image will appear in. After the 270° rotation clockwise (to the right), the image will be in quadrant 4. In quadrant 4, the x values are positive and the y values are negative.
Step 3: Switch the x and y values in each coordinate. Include the correct signs based on the location of the image. The coordinates for the image are W'(4, –9), X'(3, –5), Y'(6, –5), Z'(8, –9).
b.
Note: The identical image will be produced if quadrilateral WXYZ is rotated 90° counterclockwise.