Rectangle Problems: Bisectors And Perimeters

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Rectangle Problems: Bisectors and Perimeters

Hey guys! Let's dive into some geometry problems, specifically those involving rectangles. We'll be tackling two interesting scenarios: one involving a bisector dividing a side and another focusing on the relationship between a rectangle's sides and its perimeter. Ready to flex those math muscles?

Problem 1: The Bisector's Secret

Okay, so the first problem throws us a curveball with a bisector. Let's break it down: "The bisector of one of the angles of a rectangle divides the longer side in half. Find the perimeter of the rectangle if its shorter side is 16 cm." Sounds a bit tricky, right? Don't sweat it; we'll conquer this together! This problem tests our knowledge of rectangles, angle bisectors, and how they interact. The key here is to visualize the situation and use the properties of rectangles to our advantage. Remember, in a rectangle, all angles are 90 degrees, and opposite sides are equal. That's our foundation. The angle bisector, the line that splits an angle in half, is the hero of the problem. It forms two equal angles. Since our angle in the rectangle is 90 degrees, the bisector creates two 45-degree angles. This, combined with the right angles of the rectangle, gives us the magic key. Let's draw a picture to help us. Draw a rectangle, and label the shorter side as 16 cm. Next, draw the angle bisector from one of the corners. This bisector hits the longer side and divides it in half, which is the key piece of information. Since the angle is bisected, and we know that we have a 90 degree angle, so each of the new angles will be 45 degrees. Therefore, we can tell that the two new triangles are also right-angled, and using our trig, they have to be 45-45-90 triangles. That means our other side must also be 16 cm, as the two sides of the triangle are equal. We know that the bisector divides the longer side in half. Thus, the longer side is twice the length it is divided into. Because we've now formed a 45-45-90 triangle, we know the two sides that form this triangle are equal, and the longer side is 16cm * 2 = 32cm. Now we've got all the sides. The shorter side is given to us, and we calculate the longer side from the properties of the bisector. The perimeter is simply the sum of all sides, i.e., 2 * (16 cm + 32 cm) = 96 cm. Voila! We've cracked it. The perimeter of the rectangle is 96 cm.

Breaking Down the Solution

Here's a step-by-step breakdown of how we solved this problem to make sure everything's crystal clear:

  1. Understand the Setup: We know we're dealing with a rectangle and an angle bisector. The bisector cuts the longer side in half.
  2. Visualize and Draw: Draw a rectangle and the angle bisector. This is crucial for seeing the relationships between the sides and angles.
  3. Use Properties of Rectangles: All angles are 90 degrees, and opposite sides are equal.
  4. Angle Bisector Magic: The bisector creates two 45-degree angles.
  5. Form a Triangle and use trig: We can tell that the two new triangles are also right-angled, and using our trig, they have to be 45-45-90 triangles.
  6. Calculate the longer side: Using the properties of 45-45-90 triangles, we can use the given side and calculate the longer side.
  7. Calculate the perimeter: Use the properties of rectangles, and calculate the perimeter by adding all the sides.
  8. Final Answer: The perimeter of the rectangle is 96 cm.

See? Not so scary once you break it down! This problem highlights how understanding geometric principles and visualizing the situation can lead you to the solution.

Problem 2: Sides and Perimeter

Alright, let's switch gears and tackle the second problem: "The perimeter of a rectangle is 48 cm. Find its sides if they are in the ratio of 1:2." This one focuses on the relationship between sides and perimeter. It's all about ratios, which gives us a nice, clear path to the solution. We're given the perimeter (the total distance around the rectangle) and a ratio that describes how the sides relate to each other. In other words, one side is twice the length of the other. The challenge here is to translate this information into an equation and solve it. Let's imagine the shorter side as 'x'. Since the longer side is twice the shorter side, it's '2x'. The perimeter of a rectangle is calculated as 2 * (length + width), or 2 * (x + 2x). We know this perimeter equals 48 cm, so we can write the equation: 2 * (x + 2x) = 48 cm. Now we get to solve it! First, simplify the equation: 2 * (3x) = 48 cm. Then, 6x = 48 cm. To find the value of x, divide both sides by 6, giving us x = 8 cm. This means the shorter side is 8 cm long. Since the longer side is 2x, it's 2 * 8 cm = 16 cm. So, the sides of the rectangle are 8 cm and 16 cm. We've used ratios and the perimeter formula to determine the side lengths, guys! Keep up the good work! We're almost there! This is a great problem to show how perimeter and ratio work together! Let's think how we can get better at these problems.

Step-by-Step Breakdown

Let's break down the process step-by-step to solidify your understanding:

  1. Define Variables: Let the shorter side be 'x' and the longer side be '2x' (because of the 1:2 ratio).
  2. Use the Perimeter Formula: The perimeter of a rectangle is 2 * (length + width).
  3. Set up the Equation: 2 * (x + 2x) = 48 cm.
  4. Simplify and Solve: 6x = 48 cm, so x = 8 cm.
  5. Calculate the Sides: Shorter side = 8 cm; Longer side = 2 * 8 cm = 16 cm.
  6. Final Answer: The sides of the rectangle are 8 cm and 16 cm.

By carefully using the information given (the perimeter and the ratio), we successfully found the side lengths. Remember that clear, organized steps can make solving any problem much more straightforward.

Tips for Success

Alright, let's wrap this up with some golden nuggets of advice to help you ace these types of geometry problems.

  • Draw, Draw, Draw: Always start by drawing a diagram. It's your visual guide.
  • Label Everything: Clearly label the sides, angles, and any other relevant information on your diagram.
  • Know Your Formulas: Memorize the basic formulas for perimeters, areas, and angle properties.
  • Break It Down: Divide complex problems into smaller, more manageable steps.
  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with these concepts.
  • Understand the Vocabulary: Make sure you know what terms like