Question

given that the area of the preimage is 4, and the area of the image 1, what is the ratio of the areas? * 4 points your answer

Explanation:

Step1: Identify the areas

The area of the preimage (black triangle) is 4, and the area of the image (red triangle) is 1.

Step2: Calculate the ratio

The ratio of the areas is the area of the preimage divided by the area of the image (or vice versa, but typically preimage to image or image to preimage as per context). Here, we need the ratio of the areas, so we can take preimage area : image area or image area : preimage area. The problem says "the ratio of the areas" – usually, it's preimage to image or image to preimage. Let's check: preimage area is 4, image is 1, so ratio (preimage : image) is 4:1, or as a fraction, $\frac{4}{1}$ or $\frac{1}{4}$ depending on order. Wait, the question is "what is the ratio of the areas?" – probably preimage area to image area or image to preimage. Wait, the preimage is the original (black), image is the transformed (red). So ratio of preimage area to image area is 4:1, or image to preimage is 1:4. Wait, let's see: the problem says "the ratio of the areas" – maybe preimage over image? Wait, the preimage area is 4, image is 1. So ratio (preimage : image) is 4/1 = 4, or (image : preimage) is 1/4. Wait, let's re-read: "Given that the area of the preimage is 4, and the area of the image 1, what is the ratio of the areas?" So the ratio of the areas – probably preimage area to image area, so 4:1, or as a fraction 4/1 = 4, or image to preimage is 1/4. Wait, maybe the ratio is preimage area divided by image area, so 4/1 = 4, or image over preimage is 1/4. Wait, let's check the triangles. The black triangle: base AB, let's count the grid. AB is 2 units (from B to A: 2 squares). Height: from A to C, let's see, vertical squares: 4? Wait, no, the red triangle: base A'B' is 1 unit? Wait, AB is 2, A'B' is 1. So scale factor for length is 1/2. Then area scale factor is (1/2)^2 = 1/4. Wait, preimage area is 4, image area is 1. So 4 * (1/4) = 1. So the ratio of preimage area to image area is 4:1, and image to preimage is 1:4. But the question is "what is the ratio of the areas?" – maybe preimage to image, so 4/1 = 4, or image to preimage is 1/4. Wait, the problem says "the ratio of the areas" – maybe preimage area over image area, so 4:1, or as a fraction 4/1 = 4. Wait, let's do the calculation: ratio = area of preimage / area of image = 4 / 1 = 4, or area of image / area of preimage = 1 / 4. Which one is it? The question says "the ratio of the areas" – maybe preimage to image, so 4:1, or 4/1. Let's confirm: preimage area is 4, image is 1, so ratio (preimage : image) is 4:1, which is 4.

Answer:

The ratio of the areas (preimage area to image area) is $\boldsymbol{4:1}$ (or $\boldsymbol{\frac{4}{1}}$ or $\boldsymbol{4}$). If considering image to preimage, it's $1:4$ or $\frac{1}{4}$, but based on the problem's wording (preimage area 4, image 1), the ratio of preimage to image areas is 4.