The blue square is the preimage. Write two algebraic representations, one for the dilation to the green square and one for the dilation to the purple square.

Critical Thinking A triangle has vertices A(-5, -4), B(2, 6), and C(4, -3). The center of dilation is the origin and (x, y)→(3x, 3y). What are the vertices of the dilated image?

Critical Thinking M′N′O′P′ has vertices at M′(3, 4), N′(6, 4), O′(6, 7), and P′(3, 7). The center of dilation is the origin. MNOP has vertices at M(4.5, 6), N(9, 6), O′(9, 10.5), and P′(4.5, 10.5). What is the algebraic representation of this dilation?

Critical Thinking A dilation with center (0,0) and scale factor k is applied to a polygon. What dilation can you apply to the image to return it to the original preimage?

Represent Real-World Problems The blueprints for a new house are scaled so that \(\cfrac{1}{4}\) inch equals 1 foot. The blueprint is the preimage and the house is the dilated image. The blueprints are plotted on a coordinate plane.

a. What is the scale factor in terms of inches to inches?

b. One inch on the blueprint represents how many inches in the actual house? How many feet?

c. Write the algebraic representation of the dilation from the blueprint to the house.

d. A rectangular room has coordinates Q(2, 2), R(7, 2), S(7, 5), and T(2, 5) on the blueprint. The homeowner wants this room to be 25% larger. What are the coordinates of the new room?

e. What are the dimensions of the new room, in inches, on the blueprint? What will the dimensions of the new room be, in feet, in the new house?