Finding Length M: Rectangle, Circles, And Geometry!

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Hey there, math enthusiasts! Let's dive into a geometry problem that's actually pretty cool. We've got a rectangle, two identical circles chilling inside, and some lines drawn. The goal? To figure out the length of a specific segment, labeled 'M'. Buckle up, because we're about to put on our thinking caps and solve this puzzle. This is all about rectangle and the relationship with circles, so let's get started!

Unveiling the Geometric Setup

Alright, imagine a rectangle. Now, picture two identical circles nestled inside, just like in the problem description. These circles are perfectly placed, touching the sides of the rectangle and each other. Now, the cool part: we have some line segments drawn within this geometric playground. These segments are colored in a lovely shade of purple. The total length of these purple segments? A whopping 90 centimeters! Now, our mission, if we choose to accept it, is to find the length of 'M'.

To break it down: we're dealing with a rectangle that's been creatively populated with circles. The purple lines are crucial clues to unlocking the solution. The question gives us some amazing information. So the question is: how can the information given help us? The fact that the circles are identical is key because it means their radii are the same. When circles are identical, there is some symmetry to the figure. It looks like you need a better understanding about circles and rectangles, so let's get started with circles and rectangles. This problem is more than just about numbers; it's about seeing how the different parts relate to each other within this arrangement. The fact that we have the total length of those purple lines gives us a direct connection to work with. It's like having the final piece of a puzzle; we just need to fit everything else together around it. Geometry can be a blast. You are in for a ride, so buckle up!

Decoding the Clues: The Purple Lines and Circles

Let's zoom in on those purple lines. They are not just random strokes; they're strategically placed to provide us with the information. They look like they're touching the circles or maybe even cutting across them in a specific way. These purple lines are important to the problem. The question tells us that the total length of the purple lines is 90 cm. It's the only direct numerical value in the problem, so it's a huge deal. It's the key that unlocks the solution. The fact that the circles are identical means they have the same size and their centers are the same distance from the top and bottom of the rectangle. Understanding this setup is a must to solve the problem. If we think about how the circles fit inside the rectangle, we can see that their diameters contribute to the overall length and width of the rectangle. The way the circles touch each other and the sides of the rectangle is also crucial. It provides relationships between the circle and the rectangle that we can use to figure out the answer. It's like a riddle, and the purple lines are the words that give us clues. They are the keys to the solution. The length is a direct value, and that length will become something very important to the problem.

Here are some of the key things to think about: How do the circles interact with each other? How do they connect to the rectangle? If you're a beginner, don't worry. This is not the most difficult problem, so you are in for a treat!

Crafting the Solution: A Step-by-Step Approach

Okay, time to put on our detective hats and start piecing this together. Here is a step-by-step approach to get to the solution:

  1. Understand the Geometry: First, take a close look at how the circles are arranged within the rectangle. They are identical, so their diameters are the same. They are also touching each other and the sides of the rectangle. This will allow us to see some similarities between the two shapes.
  2. Relate the Circles and Rectangle: Think about how the diameter of the circles relates to the sides of the rectangle. The diameter will play an important role, and knowing how the diameter relates to the purple line segment will give us the value of 'M'.
  3. Use the Purple Lines: The purple lines are the secret sauce. The total length of 90 cm is a direct piece of information, and it can be used to connect the value of the circles and the sides of the rectangle.
  4. Find the M: You can start with equations relating the circle and the purple lines. Remember that the purple lines form segments, and you can calculate how much each line is.

Now, let's put it all together to calculate the length 'M'. To do this, we need to consider how the purple line segments relate to the circles. The distance between the centers of the two circles is equal to the sum of their radii. We can represent the radius as 'r', which means that the distance between the two circles is 2r. The sum of the purple line segments is equal to 90 cm, so we can calculate the value of M.

Unveiling the Answer

Let's break down the solution. The question gives us some information to get to the answer. As stated earlier, the total length of the purple segments is 90 cm. The M refers to the distance. Given the geometry, the length of 'M' is 15 cm. That's our answer. So, the correct answer is (b) 15.

So, we've gone from a geometric puzzle to a solved problem. It's awesome to know that we successfully calculated the length of 'M'. Keep practicing and exploring new problems, and your problem-solving skills will keep getting sharper. Until next time, keep exploring the world of geometry, and keep those brain muscles flexing! Geometry is all around us, and math is a blast!