Rectangle Perimeter Problems: Step-by-Step Solutions
Hey guys! Today, we're diving into some classic geometry problems involving rectangles and their perimeters. We'll break down each problem step-by-step so you can easily follow along. Let's get started!
Problem a: Finding Side Lengths from Perimeter and Ratio
So, the first problem states: The perimeter of a rectangle is 48 cm. The ratio of its sides is 1:2. Find the lengths of the sides. Let's break this down. When we see a ratio, like 1:2, it means one side is twice as long as the other. Imagine the shorter side as 'x' and the longer side as '2x'. Remember, the perimeter of a rectangle is the sum of all its sides. Therefore, in our case, that's x + 2x + x + 2x. Simplify that, and we get 6x.
Now, the problem tells us that the perimeter is 48 cm. So, we can set up the equation 6x = 48. To find 'x', we need to divide both sides of the equation by 6. That gives us x = 8. Great! We found 'x', which represents the length of the shorter side. It's 8 cm. The longer side is '2x', so that's 2 * 8 = 16 cm. So, we now know that one side of the rectangle is 8 cm and the other is 16 cm. Easy peasy!
Let's double-check our work to be absolutely sure. If the sides are 8 cm and 16 cm, the perimeter would be 8 + 16 + 8 + 16. That equals 48 cm. Bingo! We've got the right answer. Understanding the relationship between the sides of the rectangle based on their ratio and how that contributes to the overall perimeter helps in solving this kind of problem efficiently. This approach is useful in various real-world scenarios, such as designing layouts or planning constructions, where knowing the perimeter and side ratios can aid in optimizing the use of materials and space.
Problem b: Determining Perimeter Using Angle Bisector Properties
On to the next problem! It says: The bisector of angle A in rectangle ABCD intersects side BC at points dividing it into segments of 2 cm and 6 cm. Find the perimeter of the rectangle. This one's a bit trickier, but we'll tackle it together. First, let's visualize the rectangle ABCD. Angle A is bisected, meaning it's cut into two equal angles. This bisector intersects side BC, creating two segments: one is 2 cm long, and the other is 6 cm long. This tells us that the entire length of side BC is 2 + 6 = 8 cm. So, BC is 8 cm.
Now comes the cool part. Because the angle bisector creates equal angles and we know that angles in a rectangle are 90 degrees, each of the bisected angles is 45 degrees. This forms a 45-45-90 triangle. What's so special about a 45-45-90 triangle? Well, the two sides opposite the 45-degree angles are equal. In our case, this means the side adjacent to angle A (which is AB) is equal to the segment of BC that the bisector created. To clarify, if the bisector cuts BC into 2 cm and 6 cm segments and the segment closer to point B is 2 cm, then AB (another side of the rectangle) is also 8 cm. However, since the problem states that the angle bisector divides BC into segments of 2 cm and 6 cm, we need to consider the length that the bisector creates from point B. The problem states that the bisector divides BC into 2cm and 6cm, it means one part of BC is 2cm and other is 6cm.
Since we already found out that BC's length is 8cm, we need to find out the length of AB. Let's assume that the bisector intersects BC at point E. If BE is 2 cm, then AB is 2 cm. If EC is 6 cm, then we can say that BE is 6cm and AB is 6cm since ABE is a 45-45-90 triangle. Thus, two sides of our rectangle are 8 cm (BC) and 6 cm (AB). Now we can calculate the perimeter. The perimeter is 2 * (AB + BC) which is 2 * (6 + 8) = 2 * 14 = 28 cm. Thus, the perimeter of the rectangle is 28 cm. The angle bisector fundamentally changes how we approach the problem, emphasizing the importance of recognizing special triangles and their properties. Understanding these geometric relationships allows us to find missing lengths and, ultimately, the perimeter of the rectangle. This kind of problem-solving is not only useful in academic settings but also in fields like architecture and engineering where spatial reasoning and precise measurements are crucial.
Key Concepts Revisited
Let's recap what we've learned today. For the first problem, we utilized the ratio of the sides and the formula for the perimeter of a rectangle to find the individual side lengths. For the second problem, we used the properties of angle bisectors and 45-45-90 triangles to find the missing side and then calculated the perimeter. Both problems required a solid understanding of basic geometric principles and a bit of algebraic manipulation.
Final Thoughts
Geometry problems like these might seem daunting at first, but with a step-by-step approach and a good grasp of the underlying concepts, you can solve them with confidence. Keep practicing, and you'll become a geometry pro in no time! Keep these methods in mind as they can be used for different scenarios, such as maximizing the area within a given perimeter or optimizing structural designs based on specific geometric constraints. Remember, geometry isn't just about shapes and angles; it's a way of seeing and understanding the world around us. Keep exploring, keep questioning, and most importantly, keep learning! You've got this!
