Collinear Points: Finding The Length Of Segment AC

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Collinear Points: Finding the Length of Segment AC

Hey guys! Today, we're diving into a classic geometry problem involving collinear points. We've got points A, B, C, and D lined up in that order, and we're given some relationships between the lengths of the segments they form. Specifically, BC is twice the length of AB, CD is twice the length of BC, and the total length AD is 21 cm. Our mission, should we choose to accept it, is to find the length of segment AC. So, let's put on our thinking caps and get started!

Understanding the Problem

To really nail this, the very first thing we've gotta do is break down what the problem is telling us. Collinear points, as the name suggests, are points that lie on the same line. This is super important because it allows us to work with distances and segment lengths in a straightforward way. We're given the order of the points (A, B, C, D), which helps us visualize the situation. The relationships between the segment lengths (BC = 2AB, CD = 2BC) are the key to unlocking the solution. And of course, we have the total length AD = 21 cm, which is our anchor, the piece of information that ties everything together.

Think of it like a puzzle. We have these pieces of information, and we need to fit them together to reveal the length of AC. A good approach here is often to use variables to represent the unknown lengths. This allows us to translate the given relationships into algebraic equations, which we can then solve. For example, we could let the length of AB be 'x'. Then, since BC = 2AB, the length of BC would be '2x'. Similarly, we can express the length of CD in terms of x. This is a classic strategy in geometry problems, so let’s keep it in mind.

Before we jump into the algebra, it’s always a good idea to take a moment to visualize the problem. Imagine these four points lined up. A is at one end, and D is at the other. B and C are somewhere in between. The relationships between the lengths tell us that the segments are getting progressively longer as we move from A to D. BC is longer than AB, and CD is even longer than BC. This mental picture can help us catch any errors in our calculations later on, because we'll have a sense of what the answer should be in terms of relative lengths. So, visualize, visualize, visualize! It’s a powerful tool in your problem-solving arsenal.

Setting up the Equations

Okay, guys, now comes the fun part: translating the geometry into algebra! This is where we take the relationships we discussed and turn them into equations we can actually solve. Remember, we're trying to find the length of AC, so we'll need to express that in terms of our variable as well.

Let's start by assigning a variable to the length of the smallest segment, AB. It's a common and often helpful strategy to start with the smallest quantity. So, let's say AB = x. This means the length of segment AB is simply 'x' centimeters. Now, using the given relationships, we can express the lengths of the other segments in terms of x. We know that BC = 2AB, so if AB = x, then BC = 2 * x = 2x. This tells us that segment BC is twice as long as segment AB.

Next, we're given that CD = 2BC. Since we've already found that BC = 2x, we can substitute that into this equation. So, CD = 2 * (2x) = 4x. This means segment CD is four times the length of segment AB. Now we've expressed the lengths of AB, BC, and CD all in terms of x: AB = x, BC = 2x, and CD = 4x. This is a crucial step because it allows us to relate these lengths to the total length AD.

We also know that AD = 21 cm. But AD is made up of the segments AB, BC, and CD. So, we can write another equation: AD = AB + BC + CD. Now we can substitute the expressions we found for AB, BC, and CD in terms of x: 21 = x + 2x + 4x. And there you have it! We've successfully translated the geometric relationships into a single algebraic equation. This equation relates the unknown length x to the known length AD. Solving this equation will give us the value of x, which we can then use to find the length of AC. Remember, AC is the segment made up of AB and BC, so AC = AB + BC. We’re getting closer to the solution, guys! This is like building a bridge, piece by piece.

Solving for x

Alright, time to roll up our sleeves and solve for x! We've got the equation 21 = x + 2x + 4x. This is a pretty straightforward linear equation, so we can solve it using basic algebra. The first thing we want to do is simplify the equation by combining the 'x' terms on the right side. We have x + 2x + 4x, which adds up to 7x. So, our equation now becomes 21 = 7x.

Now, to isolate x, we need to get rid of the 7 that's multiplying it. We can do this by dividing both sides of the equation by 7. Remember, whatever we do to one side of the equation, we have to do to the other side to keep it balanced. So, we divide both 21 and 7x by 7: 21 / 7 = (7x) / 7. This simplifies to 3 = x. Fantastic! We've found the value of x. Remember, x represents the length of segment AB, so we now know that AB = 3 cm.

But hold on, we're not quite done yet! We're trying to find the length of AC, not just AB. We know that AC is made up of segments AB and BC, so AC = AB + BC. We already know AB = 3 cm, and we found earlier that BC = 2x. Now that we know x = 3, we can find BC: BC = 2 * 3 = 6 cm. So, segment BC is 6 cm long. Now we have all the pieces we need to find AC. We simply add the lengths of AB and BC: AC = 3 cm + 6 cm = 9 cm. Boom! We've got our answer. The length of segment AC is 9 cm.

Finding the Length of AC

Okay, so we've solved for x, found the lengths of AB and BC, and now we're ready to calculate the length of AC. As we discussed earlier, segment AC is formed by combining segments AB and BC. Therefore, to find the length of AC, we simply need to add the lengths of AB and BC together. We already determined that AB = x = 3 cm and BC = 2x = 6 cm. So, AC = AB + BC = 3 cm + 6 cm = 9 cm.

Therefore, the length of segment AC is 9 centimeters. And that’s it! We've successfully solved the problem. We started by carefully understanding the given information, setting up equations to represent the relationships between the segment lengths, solving for the unknown variable, and finally, using that information to find the length of AC. It’s like a detective story, where each step brings us closer to the solution. Remember, breaking down a problem into smaller, manageable steps is a key strategy in math and in life!

Final Answer

So, after all that awesome work, let's circle back to the original question: What is the length of segment AC? We've meticulously worked our way through the problem, setting up equations, solving for x, and finally calculating AC. And the answer, my friends, is 9 cm. Great job, guys! We tackled a geometry problem head-on and emerged victorious. Remember, the key is to break down the problem, use variables to represent unknowns, and translate the given information into equations. Keep practicing, and you'll be solving geometry puzzles like a pro in no time!