Find DC Length In Triangles ABD & DBC: A Geometry Problem

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Finding the Length of DC in Triangles ABD and DBC: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun geometry problem involving triangles, perpendicular lines, and the good ol' Pythagorean theorem. We've got triangles ABD and DBC, and our mission, should we choose to accept it, is to find the length of side DC. Let's break it down, step by step, so it's super clear and easy to follow. Are you ready to solve for x?

Understanding the Problem: Visualizing the Triangles

Before we jump into calculations, let's make sure we've got a solid picture in our minds. We have two triangles: ABD and DBC. The key piece of information here is that side BA is perpendicular to AD. What does that mean? It means we have a right angle at vertex A! This is super important because right angles open the door to using the Pythagorean theorem, our trusty tool for solving right triangles.

We know the length of AB is 8 cm, AD is 4 cm, and BC is 11 cm. Our mission is to find the length of DC, which we'll call x for simplicity. So, let's recap the givens using markdown formatting:

  • |AB| = 8 cm
  • |AD| = 4 cm
  • |BC| = 11 cm
  • |DC| = x cm (what we need to find!)

Visualizing this, we can see that triangle ABD is a right triangle, and we have the lengths of two of its sides. This means we can find the length of the third side, BD, which will be crucial for solving the bigger problem.

Step 1: Finding BD using the Pythagorean Theorem

The Pythagorean theorem is our best friend when dealing with right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In simpler terms:

  • a² + b² = c²

Where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.

In triangle ABD, BD is the hypotenuse (it's opposite the right angle at A), and AB and AD are the other two sides. So, we can plug in the values we know:

  • AB² + AD² = BD²
  • 8² + 4² = BD²
  • 64 + 16 = BD²
  • 80 = BD²

To find BD, we need to take the square root of both sides:

  • BD = √80

We can simplify √80 by factoring out the largest perfect square, which is 16:

  • BD = √(16 * 5)
  • BD = 4√5 cm

Awesome! We've found the length of BD. Now, let's move on to the next part of our adventure.

Step 2: Focusing on Triangle DBC

Now, let's shift our focus to triangle DBC. We know the length of BC (11 cm) and we've just calculated the length of BD (4√5 cm). We also want to find the length of DC (x). However, there's a slight catch: we don't know if triangle DBC is a right triangle!

To figure out x, we'll need to use the Pythagorean theorem again, but only if DBC is a right triangle. Unfortunately, we don't have any direct information about the angles in DBC. So, what do we do? Well, we assume that DBC is a right triangle with a right angle at D. Thus, we can apply the Pythagorean theorem to triangle DBC, where BC is the hypotenuse:

  • BD² + DC² = BC²

Step 3: Applying the Pythagorean Theorem to Triangle DBC

Now we can plug in the values we know for triangle DBC:

  • (4√5)² + x² = 11²
  • (16 * 5) + x² = 121
  • 80 + x² = 121

Now, let's isolate x² by subtracting 80 from both sides:

  • x² = 121 - 80
  • x² = 41

To find x, we take the square root of both sides:

  • x = √41 cm

Step 4: The Final Answer and Recap

Alright guys, we've done it! We've successfully found the length of DC. So, based on our calculations, the length of DC (x) is:

  • x = √41 cm

Let's quickly recap the steps we took to solve this geometry puzzle:

  1. Visualized the triangles: We understood the given information and drew a mental picture of triangles ABD and DBC.
  2. Found BD: We used the Pythagorean theorem in the right triangle ABD to calculate the length of BD.
  3. Focused on Triangle DBC: We shifted our attention to triangle DBC and recognized we could use the Pythagorean theorem if we assume that it is a right triangle.
  4. Applied the Pythagorean Theorem: We plugged in the values into the Pythagorean theorem equation for triangle DBC and solved for x.
  5. Final Answer: We found that x = √41 cm.

Key Takeaways and Why This Matters

This problem is a fantastic example of how the Pythagorean theorem can be applied in different scenarios. It also highlights the importance of breaking down complex problems into smaller, manageable steps. Guys, remember that geometry problems often involve combining multiple concepts and using previously calculated values to find new ones. It's like building with LEGOs – each piece fits together to create the final structure!

Why is this important? Understanding geometry is crucial for many fields, including:

  • Architecture and Engineering: Designing buildings and structures.
  • Computer Graphics and Game Development: Creating 3D models and environments.
  • Navigation and Surveying: Determining distances and locations.
  • Everyday Life: Even things like arranging furniture or figuring out the shortest route can involve geometric thinking!

So, by mastering these types of problems, you're not just acing your math class; you're building valuable skills for the future. Keep practicing, and you'll become a geometry whiz in no time! Remember, geometry is everywhere, and the more you understand it, the more you'll see the world in a whole new way.

I hope this breakdown was helpful and clear. If you have any questions or want to tackle more geometry problems together, let me know in the comments below! Keep up the awesome work, and I'll see you in the next math adventure!