a designer creates a drawing of a triangular sign on centimeter grid paper for a new business. the drawing has sides measuring 6 cm, 8 cm, and 10 cm, and angles measuring 37°, 53°, and 90°. to create the actual sign shown, the drawing must be dilated using a scale factor of 40. a. find the lengths of the sides of the actual sign. b. find the angle measures of the actual sign. c. the drawing has the hypotenuse on the bottom. the business owner would like it on the top. describe two transformations that will do this. d. the shorter leg of the drawing is currently on the left. the business owner wants it to remain on the left after the hypotenuse goes to the top. which transformation in part c will accomplish this?
The actual length is 6*40=240 cm, 8*40=320 cm, 10*40=400 cm. And the angle measures of the actual sign will not change. The transformation can be turning over using the hypotenuse as axle. Or rotate the triangular sign 180° using the center of the triangle as the center of rotation. The turning over method will accomplish.
a) To find the lengths of the sides of the actual sign, you need to apply the scale factor of 40 to each side of the original drawing.
Original side lengths: 6 cm, 8 cm, and 10 cm
Scaled side lengths: 6 cm x 40 = 240 cm, 8 cm x 40 = 320 cm, and 10 cm x 40 = 400 cm
Therefore, the lengths of the sides of the actual sign are 240 cm, 320 cm, and 400 cm.
b) To find the angle measures of the actual sign, you need to determine if the angles change during dilation. However, the angles in a triangle remain the same during dilation.
Therefore, the angle measures of the actual sign are 37°, 53°, and 90°.
c) To move the hypotenuse from the bottom to the top, you can perform two transformations:
1) Translation: Move the entire triangle vertically using the appropriate vector. For example, you can move it up by adding the same value to the y-coordinate of each vertex.
Let's say you move the triangle up by 200 cm. The new coordinates of each vertex will be:
(x, y) → (x, y+200)
2) Reflection: Reflect the triangle across the x-axis. This will flip the triangle upside down, effectively moving the hypotenuse to the top.
The new coordinates of each vertex will be:
(x, y) → (x, -y)
d) If the shorter leg of the drawing is currently on the left, and you want it to remain on the left after moving the hypotenuse to the top, the transformation required is the reflection across the y-axis.
The new coordinates of each vertex for this transformation will be:
(x, y) → (-x, y)
So, reflecting the triangle across the y-axis will accomplish this.