Collinear Points A, B, C Length Of AD Calculation

by Scholario Team 50 views

Hey guys! Let's dive into a fun geometry problem involving collinear points and square roots. We're given three points, A, B, and C, that lie on the same line. We know the distances |AB| and |AC|, and we need to figure out the possible range for the distance |AD|, where D is a point somewhere between B and C. Sounds like a plan? Let’s break it down step by step.

Understanding the Problem

So, we have these collinear points A, B, and C. Collinear just means they're all on the same line, like beads on a string. We're given the distance from A to B, which is |AB| = (√24 + √6) cm, and the distance from A to C, which is |AC| = (√54 + √96) cm. Now, imagine a point D chilling somewhere between B and C. Our mission, should we choose to accept it, is to find the possible lengths for the distance |AD|.

Breaking Down the Given Lengths

Before we jump into calculations, let's simplify those square roots. Simplifying square roots makes life so much easier, trust me! Let's start with |AB|:

  • |AB| = (√24 + √6)
  • We can rewrite √24 as √(4 * 6) which is 2√6.
  • So, |AB| = (2√6 + √6) = 3√6 cm. Awesome!

Now, let's tackle |AC|:

  • |AC| = (√54 + √96)
  • √54 can be rewritten as √(9 * 6) which is 3√6.
  • √96 can be rewritten as √(16 * 6) which is 4√6.
  • So, |AC| = (3√6 + 4√6) = 7√6 cm. Fantastic!

Visualizing the Points

Okay, so now we know |AB| = 3√6 cm and |AC| = 7√6 cm. Let's try to visualize this. Imagine a line segment starting at point A. Point B is 3√6 cm away from A, and point C is 7√6 cm away from A. Since B is closer to A than C, the order of the points on the line is A, B, then C. Make sense?

Finding the Range for |AD|

Now comes the juicy part – finding the possible lengths for |AD|. Remember, D is somewhere between B and C. This means |AD| will be longer than |AB| but shorter than |AC|. Think of it like this: if D were right on top of B, then |AD| would be the same as |AB|. If D were right on top of C, then |AD| would be the same as |AC|. But since D is somewhere in between, |AD| will be between these two lengths.

Determining the Minimum Value of |AD|

The minimum value of |AD| occurs when D is as close to B as possible. In this case, |AD| is essentially |AB|. So, the minimum value for |AD| is 3√6 cm. Easy peasy!

Determining the Maximum Value of |AD|

The maximum value of |AD| happens when D is as close to C as possible. This means |AD| is essentially |AC|. So, the maximum value for |AD| is 7√6 cm. Got it?

The Range for |AD|

So, putting it all together, the length |AD| can be anywhere between 3√6 cm and 7√6 cm. We can write this as:

  • 3√6 cm < |AD| < 7√6 cm

This means |AD| is greater than 3√6 cm but less than 7√6 cm. We use the “less than” signs because D is strictly between B and C; it can't be exactly at B or C.

Converting to a Numerical Range

For a better understanding, let's convert these lengths to approximate numerical values. We know that √6 is approximately 2.45 (you can use a calculator to find this). So:

  • 3√6 ≈ 3 * 2.45 ≈ 7.35 cm
  • 7√6 ≈ 7 * 2.45 ≈ 17.15 cm

This means |AD| is between approximately 7.35 cm and 17.15 cm. Now we have a good numerical range for the possible lengths of |AD|.

Considering Possible Answer Choices

If this were a multiple-choice question, you’d look for an answer choice that falls within this range. For example, if the choices were:

  • A) 2√6 cm
  • B) 4√6 cm
  • C) 8√6 cm
  • D) 10√6 cm

We’ve already established that |AD| must be between 3√6 cm and 7√6 cm. So, let's see which option fits:

  • A) 2√6 cm is less than 3√6 cm, so it's out.
  • B) 4√6 cm is between 3√6 cm and 7√6 cm – a potential candidate!
  • C) 8√6 cm is greater than 7√6 cm, so it's out.
  • D) 10√6 cm is also greater than 7√6 cm, so it's out.

Therefore, the correct answer would be B) 4√6 cm. See how breaking it down step-by-step makes it so much easier?

Key Takeaways

Alright, let's recap the main points we covered:

  1. Understanding Collinear Points: Collinear points lie on the same line. Visualizing them helps in solving the problem.
  2. Simplifying Square Roots: Simplifying square roots makes calculations easier and clearer. Remember to look for perfect square factors.
  3. Determining the Range: When a point is between two other points, its distance from a reference point will fall within the range of distances of the other two points.
  4. Converting to Numerical Values: Converting to numerical values can provide a better sense of scale and make it easier to compare possible answers.

Common Mistakes to Avoid

Now, let’s chat about some common pitfalls to dodge. It’s always good to know what mistakes others make so you can steer clear!

Not Simplifying Square Roots

One frequent flub is skipping the simplification of square roots. Trust me, dealing with √24 and √54 directly can be a headache. Always break down those radicals into their simplest forms. It makes calculations way smoother and reduces the risk of errors.

Misunderstanding the Range

Another typical slip-up is misunderstanding the range for |AD|. Remember, since D is between B and C, |AD| cannot be equal to |AB| or |AC|. It has to be strictly within those values. Using “less than or equal to” signs instead of “less than” can lead you astray.

Not Visualizing the Points

Failing to visualize the points on a line can also cause confusion. Drawing a quick sketch can make the relationships between the points clearer and help you understand the problem better. A picture is worth a thousand words, right?

Arithmetic Errors

Lastly, watch out for those sneaky arithmetic errors! Adding or subtracting square roots incorrectly can throw off your entire solution. Always double-check your calculations, especially when dealing with radicals.

Practice Makes Perfect

Okay, so we’ve tackled this problem head-on, simplified square roots, visualized the scenario, and even dodged some common mistakes. What’s next? Practice, practice, practice! The more you wrestle with these types of problems, the comfier you’ll get. Try tweaking the numbers, changing the distances, or even adding more points. Each twist will help solidify your understanding.

Example Problem

Let's try another one quickly. Suppose we have collinear points P, Q, and R. If |PQ| = (√18 + √8) cm and |PR| = (√50 + √32) cm, and a point S is chosen between Q and R, what could be a possible length for |PS|?

Try solving this one on your own, using the steps we discussed. Simplify the square roots, determine the range for |PS|, and then check if your answer makes sense. Remember, you've got this!

Conclusion

So, there you have it, folks! We’ve conquered a collinear points problem, simplified radicals, found a range for distances, and even talked about common mistakes to avoid. Geometry problems like these might seem daunting at first, but with a systematic approach and a sprinkle of visualization, they become totally manageable. Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys are awesome, and you've totally got this. Until next time, happy problem-solving!