Dilating A Pentagon: Find S' Coordinates

Dec 16, 2025 by Andrew McMorgan 41 views

Dilating a Pentagon: Find S' Coordinates

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of geometry and specifically tackling a question that might make some of you scratch your heads: What is the ordered pair of point S' after dilating pentagon OPQRS by a scale factor of 7/4 from the origin? Don't worry, we're going to break this down step-by-step, making it super clear and easy to understand. So, grab your notebooks, get comfy, and let's get started on this awesome mathematics adventure!

Understanding Dilation in Geometry

First things first, let's get our heads around what dilation actually means in geometry. Think of it like using a photocopier or a zoom lens. When you dilate a shape, you're essentially making it bigger or smaller, but you're keeping its shape the same. It's like stretching or shrinking a rubber band – the shape stays consistent, but the size changes. The key player here is the scale factor. This number tells us how much we're scaling. A scale factor greater than 1 makes the shape bigger, while a scale factor between 0 and 1 makes it smaller. If the scale factor is negative, the shape is also reflected across the origin. In our specific problem, the scale factor is $\frac{7}{4}$ , which is greater than 1, so we know our new pentagon, O'P'Q'R'S', will be larger than the original pentagon OPQRS. The center of dilation is also super important; it's the point from which the dilation happens. In this case, our center of dilation is the origin (0,0). This means every point in the original pentagon is stretched or shrunk directly away from or towards the origin.

To perform a dilation from the origin with a scale factor 'k', you simply multiply the x-coordinate and the y-coordinate of each point by 'k'. So, if you have a point (x, y), its image after dilation will be (kx, ky). This is the fundamental rule we'll be using to find the coordinates of S'. Remember, this applies to every single point of the shape, not just the vertices. While the problem specifically asks for point S', understanding this rule for all points is crucial for a complete grasp of dilation. It's this consistent application of the scale factor across all coordinates that preserves the similarity between the original and dilated shapes. The angles remain the same, and the ratios of corresponding side lengths are all equal to the scale factor.

The Importance of the Origin as the Center of Dilation

Now, let's talk about why the origin being the center of dilation is a big deal. When the center of dilation is the origin (0,0), the math becomes incredibly straightforward. As we just discussed, you just multiply each coordinate of a point by the scale factor to find the coordinates of its image. This simplicity makes problems like the one we're solving today much more manageable. If the center of dilation were a different point, say (a,b), the process would be a bit more involved. You'd first have to translate the shape so that the center of dilation moves to the origin, then perform the dilation, and finally translate the shape back. This involves a few more steps and can be a bit trickier. So, when you see that the dilation is from the origin, give yourself a little pat on the back because it's the easiest scenario!

For instance, if we had a point (x, y) and the center of dilation was the origin (0,0) with a scale factor 'k', the new point (x', y') would be calculated as: $x' = kx$ and $y' = ky$ . This is a direct multiplication. However, if the center of dilation was (a,b), the process would look like this:

Translate the point (x,y) so the center of dilation (a,b) moves to the origin. The translated point becomes (x-a, y-b).
Dilate this translated point by the scale factor 'k'. The dilated point is $(k(x-a), k(y-b))$ .
Translate the dilated point back by adding the original center coordinates (a,b). The final image point (x', y') would be $(k(x-a) + a, k(y-b) + b)$ .

See? It's definitely more complex. Therefore, identifying the center of dilation is a critical first step in any dilation problem. Since our problem explicitly states the dilation is from the origin, we can stick to the simpler multiplication rule. This is a classic example of how understanding the foundational concepts in mathematics can significantly simplify problem-solving. It highlights the elegance and interconnectedness of geometric transformations.

Finding the Coordinates of S'

Alright guys, the moment we've all been waiting for! We need to find the ordered pair of point S'. The problem states that pentagon OPQRS is dilated by a scale factor of $\frac{7}{4}$ from the origin to create O'P'Q'R'S'. To find the coordinates of S', we need the original coordinates of point S. Unfortunately, the problem doesn't give us the specific coordinates of point S. This is a common setup in math problems, designed to test your understanding of the process rather than just plugging in numbers. So, let's assume point S has coordinates $(x_S, y_S)$ .

Using the rule for dilation from the origin with a scale factor $k = \frac{7}{4}$ , the coordinates of the image point S' will be $(k imes x_S, k imes y_S)$ . Substituting our scale factor, the coordinates of S' will be $(\frac{7}{4} imes x_S, \frac{7}{4} imes y_S)$ .

So, the ordered pair of point S' is $(\frac{7}{4}x_S, \frac{7}{4}y_S)$ .

To give you a concrete example, let's imagine that the original point S had coordinates (4, 8). If we dilate this point from the origin by a scale factor of $\frac{7}{4}$ , the new coordinates for S' would be:

$x_{S'} = \frac{7}{4} imes 4 = 7$

$y_{S'} = \frac{7}{4} imes 8 = 7 imes 2 = 14$

So, in this hypothetical case, S' would be at the point (7, 14). The key takeaway here is that no matter what the original coordinates of S are, the process to find S' remains the same: multiply the original x-coordinate by $\frac{7}{4}$ and the original y-coordinate by $\frac{7}{4}$ . This method ensures that the relationship between the original point and its dilated image is maintained according to the scale factor and the center of dilation. It's a fundamental concept in understanding how geometric transformations affect points in a coordinate plane. The preservation of this proportional relationship is what defines dilation and ensures the resulting shape is similar to the original.