How can you determine the coordinates of any image that is dilated with the center of dilation at the origin without graphing? Explain your reasoning.

All lengths grow by the same factor, so

just multiply every coordinate by the same value.

Well, trying to find the coordinates of a dilated image without graphing is like trying to find your way out of a maze blindfolded while juggling bowling pins – it's not gonna be easy! But fear not, I'm here to help you navigate this mathematical conundrum with a touch of humor.

So, when we talk about dilation, we basically mean stretching or shrinking an image. When the center of dilation is at the origin, things get a bit simpler. Think of the origin as the base camp where all the action begins.

To determine the coordinates of a dilated image, you need to remember that dilation involves multiplying the original coordinates by a scale factor. This scale factor determines the amount of stretching or shrinking the image undergoes.

Let's say the original coordinates of a point are (x, y). The dilated coordinates would be given by (kx, ky), where k is the scale factor. But here's the catch – you still need to know the value of k! Without graphing or any additional information, determining the exact scale factor can be as tricky as trying to tickle a gorilla without getting squashed.

So, to summarize, to find the coordinates of a dilated image with the center of dilation at the origin without graphing, you'll need to know the scale factor. But if you're feeling as clueless as a fish on a bicycle, it might be a good idea to grab a graph and some colored pencils to help you out!

To determine the coordinates of an image that is dilated with the center of dilation at the origin without graphing, you can follow these steps:

1. Determine the scale factor (or dilation factor) of the dilation. This is given by the ratio of the lengths of corresponding sides or distances between the original point and the dilated point.

2. For each point in the original image, multiply its coordinates (x, y) by the scale factor. This will give you the new coordinates (x', y') of the dilated image.

3. Repeat step 2 for each point in the original image.

This reasoning is based on the properties of dilations. When a figure is dilated with the center of dilation at the origin, all distances from the origin are multiplied by the same factor. Therefore, multiplying the coordinates of each point by the scale factor will give you the coordinates of the dilated image. This method allows you to determine the coordinates without the need for graphing the image.

To determine the coordinates of an image that is dilated with the center of dilation at the origin without graphing, you can use the formula for dilation.

First, let's understand the concept of dilation. Dilation is a transformation that expands or contracts an object by a scale factor, which is the ratio of the length of the image to the length of the pre-image.

For any point (x, y) in the pre-image, the coordinates of the dilated point (x', y') can be found using the formula:
x' = k * x
y' = k * y

Where 'k' is the scale factor.

In this case, since the center of dilation is at the origin (0, 0), the formula can be simplified to:
x' = k * x
y' = k * y

To determine the coordinates of the dilated image, you need to know the scale factor 'k' and the coordinates of the pre-image's points.

For example, let's say the scale factor 'k' is 2, and we have a pre-image point (2, 3).
To find the corresponding dilated image point, substitute the values into the formula:
x' = 2 * 2 = 4
y' = 2 * 3 = 6

So, the dilated image point would be (4, 6).

By applying this formula to each point of the pre-image, you can determine the coordinates of the dilated image without graphing. Just remember to use the same scale factor 'k' for each point.