Homothety And Area Reduction: Solving The Photo Problem

Hey there, math enthusiasts! Today, we're diving into a cool geometry concept called homothety, and we'll use it to solve a fun problem about shrinking photographs. Imagine you have a rectangular photo, and you want to make a smaller version of it. Homothety is the perfect tool for this! Let's break down the problem step-by-step and see how it all works. We'll explain what homothety is, how it affects the area of a shape, and then apply our knowledge to solve the problem about Leninha and her photo. So, buckle up, and let's get started!

Understanding Homothety: The Geometry of Scaling

So, what exactly is homothety? Think of it as a geometric transformation that changes the size of a figure without altering its shape. It's like using a copy machine to enlarge or reduce a picture. Homothety is defined by two key elements: a center of homothety (a fixed point) and a scale factor (also called the ratio of homothety). The center of homothety is the point around which the figure is scaled. The scale factor determines how much the figure is enlarged or reduced. If the scale factor is greater than 1, the figure is enlarged; if it's between 0 and 1, the figure is reduced; and if it's negative, the figure is flipped and scaled.

Let's illustrate this with an example. Imagine you have a triangle. If you apply a homothety with a scale factor of 2, the new triangle will have sides twice as long as the original. Its area, however, will be four times as big (because area scales with the square of the side length). If you use a scale factor of 1/2, the new triangle will have sides half as long, and its area will be one-fourth of the original. Homothety is a powerful concept because it preserves the shape of the original figure. The new figure is a scaled version of the original, with the same angles and proportions. This property makes homothety useful in various fields, including computer graphics, architecture, and even art! Understanding how the scale factor affects the area is crucial. The area of the new figure is equal to the area of the original figure multiplied by the square of the scale factor. This is a critical concept that we'll use to solve Leninha's photo problem. Always keep this in mind: when scaling by a factor k, the area scales by a factor of k².

Leninha's Photo Problem: A Step-by-Step Solution

Alright, guys, let's get down to the nitty-gritty of Leninha's photo problem! Leninha used homothety to reduce a rectangular photo ABCD to a smaller photo EFGH. The original photo's area was 90 cm², and the ratio of homothety was 1/3. The question is: what is the area of the reduced photo EFGH? Let's break this down step-by-step to make sure we've got it clear.

We know the original area (90 cm²) and the scale factor (1/3). The most important thing here is to remember how the area changes with homothety. As we discussed earlier, the area of the new figure is equal to the original area multiplied by the square of the scale factor. Here's the formula: New Area = Original Area × (Scale Factor)².

Now, let's plug in the numbers: Original Area = 90 cm², Scale Factor = 1/3. So, New Area = 90 cm² × (1/3)². Calculating (1/3)² gives us 1/9. Therefore, New Area = 90 cm² × (1/9). Now, we just need to perform the multiplication: 90 × (1/9) = 10 cm². That means the area of the reduced photo EFGH is 10 cm²! Easy peasy, right? So, the correct answer to the problem is (A) 10 cm². Homothety makes the calculation super straightforward. All you need to remember is how the area scales based on the scale factor. By understanding this concept, you can solve similar problems involving area reduction or enlargement with ease. You're equipped to handle any problem involving homothety and area calculations. Keep practicing, and you'll master these concepts in no time. Math can be really fun once you get the hang of it!

Applying the Homothety Formula: A Comprehensive Example

Let's walk through another example to cement your understanding of how homothety affects area. Suppose we have a square with a side length of 6 cm. Therefore, the original area of this square is 6 cm * 6 cm = 36 cm². We then apply a homothety with a scale factor of 1/2. So, what is the area of the new square? First, identify the scale factor which is 1/2. Next, remember that the area changes by the square of the scale factor. Therefore, the new area equals the original area multiplied by (1/2)². (1/2)² = 1/4. Then we compute the new area: 36 cm² * (1/4) = 9 cm². The new square has an area of 9 cm². This makes sense because the sides are now half the original size, and the area is one-fourth of the original. If we had increased the original square using a scale factor of 2, then the area would have been multiplied by 2² or 4, resulting in an area of 144 cm². See? The scale factor always affects the area by its square value! Another crucial concept to understand is that the center of homothety determines the fixed point for the scaling. Imagine that the center of homothety is at one corner of the original square. After applying the homothety, that corner will remain at its original position. The other corners will move closer to the center by the scale factor. This means that the shape shrinks or expands in relation to the center of homothety. Let's suppose we have a triangle, and the center of homothety is inside the triangle. Then, the homothety will either shrink or expand the triangle, keeping the shape similar but changing its size. Always make sure you know the scale factor, the center of homothety, and the original area. With those, you can figure out the final area with ease!

Why Homothety Matters: Practical Applications

So, why should you care about homothety? Well, it's more than just a fun math concept; it has real-world applications. In architecture and design, homothety is used to create scaled models of buildings. Imagine designing a skyscraper. You'd first build a smaller model to test its design. That model would be a homothety of the actual building. In computer graphics, homothety is used to resize images and create effects. When you zoom in or out on a picture, the software is essentially performing a homothety transformation. In art, artists use homothety to create perspective and achieve realistic representations of objects. The ability to scale objects proportionally is essential to achieving accurate perspectives. It's also used in mapmaking to scale maps, preserving the shapes of geographical features. Basically, homothety is a tool that allows us to change the size of things while maintaining their shape. Pretty cool, huh? The ability to understand and apply these concepts is useful in many situations! It can come in handy in everyday life, even if you don't realize it. Homothety also ties in with other geometric concepts, like similarity and congruence. Two figures that are related by homothety are similar, meaning they have the same shape but different sizes. Understanding homothety helps build a solid foundation for further study in geometry and related fields. So, keep exploring and practicing. The more you work with it, the easier it becomes to grasp and apply!