Cube Distance: Point C To Midpoint Of AB

by TextBrain Team 41 views

Let's dive into a fun geometry problem involving a cube! We're given a cube named ABCD.EFGH, and each of its sides (or edges) is 6 cm long. The challenge? Find the distance between point C and the midpoint of line segment AB. Sounds like a plan? Let's break it down step by step so everyone can follow along, and you'll see it's not as scary as it might seem at first.

Visualizing the Cube

Before we start crunching numbers, it's super helpful to visualize what we're dealing with. Imagine a cube – think of a dice or a perfectly square box. Label the corners of one face as ABCD, going in order around the square. The top face would then be EFGH, sitting right above ABCD, with E above A, F above B, G above C, and H above D. Got that picture in your head? Great! Now, find the middle of the line AB – that's the midpoint we're interested in.

Visualizing the cube is important.

Setting Up the Problem

Okay, now that we have our cube in mind, let's set up the problem. We know the side length (AB) is 6 cm. Let's call the midpoint of AB point M. We need to find the distance CM. To do this, we can use the Pythagorean theorem, which is a powerful tool for solving distance problems in geometry. We're going to create a right triangle using points C, M, and another strategic point on the cube. This will make it easier to calculate the distance CM.

Finding the Distance

To find the distance from point C to the midpoint M of AB, we'll use the Pythagorean theorem. Here's how we'll set it up:

  1. Identify the Right Triangle: Consider triangle CMB. However, this isn't a right triangle, so we can't directly apply the Pythagorean theorem. Instead, let's consider point A. Now, look at the triangle AMC. Still not a right triangle. Let's think a bit outside the box here.

  2. Using Coordinates: Imagine placing the cube in a 3D coordinate system. Let A be at the origin (0,0,0). Then B is at (6,0,0), C is at (6,6,0), and D is at (0,6,0). Since M is the midpoint of AB, its coordinates are (3,0,0).

  3. Calculate the Distance: Now we can use the distance formula to find the distance CM. The distance formula in 3D is:

    √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

    In our case, C is (6,6,0) and M is (3,0,0). Plugging these values into the formula:

    CM = √[(6 - 3)² + (6 - 0)² + (0 - 0)²]

    CM = √[(3)² + (6)² + (0)²]

    CM = √[9 + 36 + 0]

    CM = √45

    CM = √(9 * 5)

    CM = 3√5 cm

So, the distance between point C and the midpoint of AB is 3√5 cm. That's the final answer!

Alternative Approach: Using a 2D Plane

Okay, so the coordinate method works great, but let's explore another way to solve this, focusing on a 2D plane. This approach might feel a bit more intuitive for some folks. Let's break it down:

  1. Visualize the Setup: Imagine the square ABCD lying flat. We want to find the distance from C to M, where M is the midpoint of AB. This forms a triangle AMC, but as we noted earlier, it's not a right triangle. So, we need to get creative.

  2. Create a Right Triangle: Draw a line from C perpendicular to the line AB extended. Let's call the point where this perpendicular line meets the extended AB line point N. Now, we have a right triangle CNC.

  3. Determine Lengths:

    • AM = MB = 3 cm (since M is the midpoint of AB)
    • BC = 6 cm (side of the cube)
    • Since we extended AB to meet the perpendicular from C, we can see that AN = AB + BN. Because CN is perpendicular to AB extended, and angle CBN is a right angle (90 degrees) and angle BCN is 0 degrees. and angle CNB is a right angle (90 degrees) and length BC is 6, this makes BN equal to 0, making AN = AB = 6.
    • Now, look at the right triangle CNB. CN (the vertical distance from C to the extended AB) is equal to BC which is 6cm.
    • Triangle MNC, length MN = AM + AN. MN = 3 + 0. MN = 3cm.

  4. Apply the Pythagorean Theorem:

    • In right triangle CMN, we have:

      CM² = MN² + CN²

      CM² = (3)² + (6)²

      CM² = 9 + 36

      CM² = 45

      CM = √45

      CM = 3√5 cm

So, using this method, we arrive at the same answer: the distance from point C to the midpoint M of AB is 3√5 cm.

Conclusion

We've successfully found the distance between point C and the midpoint of AB in the cube ABCD.EFGH. We used two different approaches: the coordinate method and creating a right triangle in a 2D plane. Both methods led us to the same answer: 3√5 cm. Geometry problems can be a lot of fun once you break them down into smaller, manageable steps!

Remember, guys, the key to solving these problems is visualization and understanding the properties of the shapes you're working with. Don't be afraid to draw diagrams and try different approaches until you find one that clicks. Keep practicing, and you'll become a geometry whiz in no time!

So the answer is (d) 3\[sqrt53\[sqrt{5} cm. Keep up the great work!