Calculating Shaded Area: A Step-by-Step Guide

Hey guys! Let's dive into a fun geometry problem where we'll calculate the area of a shaded region. This problem involves a square, some equal segments, and a little bit of clever thinking. Ready to get started? Here's the deal: We've got a square called ADFG with a side length of 12b. Inside this square, we have some points that divide the sides into equal parts, creating a shaded region we need to find the area of. The problem tells us that AB = BC = CD and AH = HG = DE = EF. Let's break down how to solve this step-by-step.

Understanding the Problem and Setting Up

First things first, let's make sure we totally understand what we're dealing with. We have a square, which means all four sides are equal, and all the angles are right angles (90 degrees). Knowing the side length of the square, 12b, is crucial. Because AB = BC = CD, we know that the side AD is divided into three equal segments. Therefore, AB = BC = CD = (12b) / 3 = 4b. Similarly, since AH = HG = DE = EF, the side AF is also divided into three equal segments, meaning AH = HG = DE = EF = (12b) / 3 = 4b. It's like we're slicing the square into smaller parts with these equal segments. The shaded area is the area we want to find, which is the trickiest part of this puzzle. We need to find the area of the shaded region to achieve the final result, which is why we have to use the information given. This is our first step: understanding all the variables. Remember that the sides of the square are our main variables. The areas are the result of a calculation, so we have to take into account all of the data given to us.

Now, let's consider what shapes make up the shaded region. Looks like it's a combination of triangles and other quadrilaterals. Remember that the area of a triangle is calculated by the following formula: (1/2) * base * height. Also, keep in mind that we can use the properties of squares and the equal segments to find the lengths of the bases and heights of these shapes. With this information, we're ready to proceed, which will help us in the next step. The most important thing is to be sure that you have your problem set up and understand the main variables. Without this, you will not be able to solve the problem, and you'll be lost in the calculations.

Finding the Area of the Unshaded Triangles

Now, let's go for the kill. Finding the area of the unshaded regions is a strategic move to help us. Focusing on the unshaded triangles will lead us to the final solution. We're going to subtract the area of the unshaded region from the total area of the square. We'll calculate the areas of the unshaded triangles first. Notice that we have four triangles in the corners of the square. Each one is a right triangle because the corners of a square are right angles. Because we have the data of all the sides, we can calculate this quite easily. The most important thing is to understand what the variables mean and how they work.

Let's focus on triangle ABH. The base AB is 4b, and the height AH is also 4b. The area of triangle ABH is (1/2) * base * height = (1/2) * (4b) * (4b) = 8b². Next, consider the triangle CDE. The base CD is 4b, and the height DE is also 4b. The area of triangle CDE is (1/2) * (4b) * (4b) = 8b². Since we also have the data of the other sides, we can then calculate the total area of the two triangles. The sum of the areas of triangles ABH and CDE is 8b² + 8b² = 16b². Let's not forget about the other two triangles. The other two triangles are the same, so it will be easier to calculate. The key is to be methodical. By calculating each triangle separately, we avoid confusion and increase the chances of getting the right solution. Remember to double-check your calculations and units. By going step-by-step, you'll get the correct result. The main trick is to divide and conquer. You have to calculate the areas of the triangles, and then you can proceed to the next step.

Calculating the Total Area and Shaded Area

Now, we have to calculate the total area of the square ADFG. The area of a square is side * side, so the total area is (12b) * (12b) = 144b². Now comes the exciting part. To find the shaded area, we have to subtract the areas of the unshaded triangles from the total area of the square. Remember, we already calculated the total area of the unshaded triangles, which is 16b². Let's start with that. The area of the square is 144b², and we subtract the area of the two triangles. The area of the other two triangles, BCG and FEG, can be determined in the same way. The base BC is 4b, and the height CG is 8b. The area of triangle BCG is (1/2) * (4b) * (8b) = 16b². The same applies for triangle FEG, where the area will also be 16b². The total area of the four unshaded triangles is 16b² + 16b² + 16b² + 16b² = 64b². Remember that this is the key to solving the problem. We now have to subtract this from the total area. The shaded area is 144b² - 64b² = 80b². This result means we successfully calculated the area of the shaded region.

This is the final result, and with this information, we completed the exercise. That's it! We've successfully calculated the area of the shaded region. With a bit of careful thinking and breaking the problem down step by step, we found the answer. Remember to always double-check your work and make sure your calculations are correct. Understanding the properties of shapes like squares and triangles, and knowing the formulas for their areas, are essential tools for solving geometry problems. Keep practicing, guys, and you'll get better at it with each problem you solve! It's all about breaking down complex problems into smaller, manageable steps. Congratulations, you are now ready for the next challenge!