91Ó°ÊÓ

Complete Parts a–d. a. Find the coordinates of each vertex, and copy the figure onto graph paper. b. Perform the given rule on the coordinates of each vertex to find the image vertices. c. On the same set of axes, plot each image point. Connect them in order. d. Compare the image to the original: is it a reflection, a rotation, a translation, or some other transformation? Rule: \((x, y) \rightarrow(x+2, y-3)\)

Complete Parts a–d. a. Find the coordinates of each vertex, and copy the figure onto graph paper. b. Perform the given rule on the coordinates of each vertex to find the image vertices. c. On the same set of axes, plot each image point. Connect them in order. d. Compare the image to the original: is it a reflection, a rotation, a translation, or some other transformation? Rule: \((x, y) \rightarrow(x-1, y-1)\)

Do your work for Exercises 6 and 7 without using tracing paper. Draw a quadrilateral \(A B C D .\) Choose a point to be the center of rotation, and rotate your quadrilateral 80° about that point. Use prime notation to label the image vertices. For example, the image of vertex \(A\) will be vertex \(A^{\prime}.\)

Consider what happens when you reflect a linear graph. a. Graph the line \(y=2.5 x+4\) b. On the same axes, draw the image of this line after reflection over the x-axis. c. Write an equation of the new line. d. On the same axes, draw the image of the original line after reflection over the y-axis. e. Write an equation of the new line. f. What do you notice about the two image lines you drew? g. Do your equations in Parts c and e support your observation in Part f? Explain.

Consider what happens when you rotate a linear graph \(180^{\circ} .\) a. Graph the line \(y=2 x+4\) b. On the same grid, draw the image of the line under a \(180^{\circ}\) rotation centered at the origin. c. What do you notice about the image line and the original line? d. Write an equation of the new line. e. Does your equation in Part d support your observation in Part c? Explain.

Perspective drawings look three-dimensional. The projection method for making scale drawings is related to a method for making perspective drawings. On your own paper, follow the steps below to make a perspective drawing of a box. Use a pencil. a. Start by drawing a rectangle. This will be the front of your box. b. Choose a point outside your rectangle. This point is called the vanishing point for your drawing. Connect each vertex to that point, and then find the midpoint of each connecting segment. c. Connect the four midpoints you found in Part b to each other, in order. This gives you the back of the box. Then erase the lines connecting them to the vanishing point. d. To make the box clearer, erase the lines that should be hidden on the back of the box, or make them dashed. e. Follow the same steps to make a perspective drawing of a triangular prism. That is, start with a triangle (instead of a rectangle) and follow Part a–d. f. In this method of three-dimensional drawing, at what step do you create a pair of similar figures? Explain.

If the angle of rotation for a figure with rotation symmetry is an integer, it is also a factor of 360. Consider what might happen if you tried to create a figure using an angle measure that isn’t a factor of 360, such as 135°. Choose a point A and a center of rotation. Rotate Point A 135°, and rotate the image 135°. Keep rotating the images until you return to the original point. (When you perform the rotations, you will pass the original point, because you have made one full turn.) a. How many full circles did you make? b. How many copies of the point do you have in your drawing? c. There is an angle of rotation smaller than 135° that you could have used to create this same design. What is its measure? d. Now find the greatest common factor (GCF) of 135 and 360. e. Divide 135 and 360 by your answer to Part d. f. Compare your answers for Parts a–c to your answers for Parts d and e. What do you notice? g. Suppose you created a figure by rotating a basic design element 80° each time. What angle of rotation will the final design have? Test your answer by rotating a single point.

Similar to lines of symmetry for two-dimensional figures, three- dimensional objects can have planes of symmetry. For example, a cube has nine planes of symmetry, including these three: a. How many planes of symmetry does a regular square pyramid have? Describe or sketch them. b. How many planes of symmetry does a sphere have? Describe or sketch them. c. How many planes of symmetry does a right hexagonal prism have? Describe or sketch them.

Find the value of \(t\) in each equation. \(t^{5}=32\)

Evaluate each expression for a 2 and b 3. $$ 4^{a} $$