Question

preimage image scale factor = 1/4 ? blue rectangle blue rectangle with 5 cm and a = 20 cm² given the area, what is the length of the missing side in the preimage? ∘ 4 cm ∘ 320 cm ∘ 16 cm ∘ 20 cm

Explanation:

Step1: Find the height of the image rectangle

The area of a rectangle is \( A = \text{length} \times \text{height} \). For the image, \( A = 20 \, \text{cm}^2 \) and length \( = 5 \, \text{cm} \). So, \( \text{height} = \frac{A}{\text{length}} = \frac{20}{5} = 4 \, \text{cm} \).

Step2: Relate image and preimage using scale factor

The scale factor from preimage to image is \( \frac{1}{4} \), meaning the image dimensions are \( \frac{1}{4} \) of the preimage. Let the preimage height be \( h \). Then \( 4 = \frac{1}{4}h \), so \( h = 4 \times 4 = 16 \, \text{cm} \)? Wait, no—wait, the missing side in preimage: wait, the image length is 5 cm, scale factor \( \frac{1}{4} \), so preimage length? Wait, no, the image has length 5 cm, area 20, so height 4. The scale factor is \( \frac{1}{4} \), so preimage dimensions are 4 times image. Wait, maybe the missing side is the height? Wait, no, let's re-express.

Wait, the image is a rectangle with length 5 cm, area 20 cm². So image height \( h_i = \frac{20}{5} = 4 \, \text{cm} \). The scale factor from preimage to image is \( \frac{1}{4} \), so preimage height \( h_p = h_i \div \frac{1}{4} = 4 \times 4 = 16 \, \text{cm} \)? Wait, no, scale factor \( k = \frac{1}{4} \), so image = preimage \( \times k \). So preimage = image \( \div k \). So if image length is 5, preimage length is \( 5 \div \frac{1}{4} = 20 \)? Wait, no, maybe the missing side is the height? Wait, no, the options include 20. Wait, let's check again.

Wait, image: length 5, area 20, so height 4. Scale factor \( \frac{1}{4} \), so preimage dimensions are 4 times image. So preimage length: \( 5 \times 4 = 20 \)? Wait, no, scale factor is from preimage to image, so image = preimage (1/4). So preimage = image 4. So image length is 5, so preimage length is 54=20. Image height is 4, preimage height is 44=16. But the question is about the missing side in the preimage. Wait, the preimage has a missing side (the height? Or length?). Wait, the image's length is 5, which is 1/4 of preimage's length. So preimage length is 5 / (1/4) = 20. Wait, maybe the missing side is the length. Let's check the options: 20 cm is an option.

Wait, let's redo:

1. Image rectangle: length \( l_i = 5 \, \text{cm} \), area \( A_i = 20 \, \text{cm}^2 \). So height \( h_i = \frac{A_i}{l_i} = \frac{20}{5} = 4 \, \text{cm} \).

1. Scale factor \( k = \frac{1}{4} \) (preimage to image). So for linear dimensions, \( l_i = k \times l_p \), so \( l_p = \frac{l_i}{k} = \frac{5}{\frac{1}{4}} = 20 \, \text{cm} \). Similarly, \( h_i = k \times h_p \), so \( h_p = \frac{h_i}{k} = \frac{4}{\frac{1}{4}} = 16 \, \text{cm} \). But the question is about the missing side in the preimage. The preimage's side (the one corresponding to the image's length 5) would be 20. So the missing side is 20 cm.

Answer:

20 cm