Cube Geometry: Side Length, Diagonal, And Distance Calculations

Let's dive into the fascinating world of cube geometry! We're going to explore a cube named STUV.WXYZ and dissect some interesting properties related to its side length, diagonals, and distances between specific points. This should be fun, guys! We'll be focusing on understanding how to calculate these properties given some initial information.

Understanding the Cube

Before we jump into the statements, let’s visualize our cube STUV.WXYZ. A cube, by definition, is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Imagine a perfect dice; that’s essentially what we are dealing with. All sides are equal in length, and all angles are right angles. The beauty of a cube lies in its symmetry and predictability. Understanding the relationships between its edges, face diagonals, and space diagonals is key to solving many geometry problems. Remember that the face diagonal is the diagonal of one of the square faces, while the space diagonal connects opposite corners of the cube through its interior.

In this case, we are given that the face diagonal is $3\sqrt{8}$. This is our starting point, our known value. We will use this piece of information to verify the subsequent statements about the cube. We'll need to recall some fundamental geometric principles, such as the Pythagorean theorem, to relate the face diagonal to the side length of the cube. From there, we can deduce other distances and lengths within the cube, like the length of TZ (a space diagonal) and the distance from a vertex to the midpoint of an edge. It's all about connecting the dots! So, with our cube firmly in mind and our geometric tools at the ready, let's examine the first statement.

Statement 1: The Length of the Cube's Edge

The first statement claims that the length of the cube's edge is 3 cm. To verify this, we need to use the information about the face diagonal, which is given as $3\sqrt{8}$ cm. Remember, the face of a cube is a square, and the diagonal of a square divides it into two right-angled triangles. The sides of the square are the legs of the right triangle, and the face diagonal is the hypotenuse. Here comes the Pythagorean theorem!

Let's denote the side length of the cube as 's'. According to the Pythagorean theorem: $s^2 + s^2 = (3\sqrt{8})^2$. Simplifying this equation, we get $2s^2 = 9 * 8 = 72$. Dividing both sides by 2, we have $s^2 = 36$. Taking the square root of both sides, we find that $s = 6$. Therefore, the length of the cube's edge is actually 6 cm, not 3 cm. Thus, statement 1 is incorrect. It's essential to be precise with these calculations, as a small mistake can lead to a wrong conclusion. So, we've debunked the first statement. Let's move on to the next one, keeping in mind the correct side length we just calculated.

Statement 2: The Length of TZ

Statement 2 proposes that the length of TZ is $6\sqrt{3}$ cm. Now, what is TZ? Looking back at our cube STUV.WXYZ, TZ represents a space diagonal of the cube. A space diagonal connects two opposite vertices of the cube, passing through its interior. To find the length of a space diagonal, we can use the formula $d = s\sqrt{3}$, where 'd' is the length of the space diagonal and 's' is the side length of the cube. We already determined that the side length 's' is 6 cm.

Plugging this value into the formula, we get $d = 6\sqrt{3}$ cm. Wow, it matches! This confirms that the length of TZ, the space diagonal, is indeed $6\sqrt{3}$ cm. Therefore, statement 2 is correct. This demonstrates the power of knowing the formula for the space diagonal and accurately calculating the side length. It's like having a secret code to unlock the cube's hidden dimensions. Now, let’s tackle the third statement and see if it holds up.

Statement 3: The Distance from S to the Midpoint of TU

The third statement claims that the distance from point S to the midpoint of TU is $3\sqrt{5}$ cm. Let's visualize this. Point S is a vertex of the cube, and the midpoint of TU is located on one of the cube's edges. Let's call this midpoint M. We need to find the length of the line segment SM. To do this, we can use the Pythagorean theorem in a slightly more complex way. Imagine a right triangle formed by the points S, M, and T. The length of ST is a side of the cube, which we know is 6 cm. The length of TM is half the length of TU, since M is the midpoint. Therefore, TM = 6/2 = 3 cm.

Now, we can use the Pythagorean theorem to find SM: $SM^2 = ST^2 + TM^2$. Plugging in the values, we get $SM^2 = 6^2 + 3^2 = 36 + 9 = 45$. Taking the square root of both sides, we find that $SM = \sqrt{45} = \sqrt{9 * 5} = 3\sqrt{5}$ cm. Bingo! The distance from point S to the midpoint of TU is indeed $3\sqrt{5}$ cm. Therefore, statement 3 is correct. This calculation reinforces the importance of visualizing the geometry and breaking down the problem into smaller, manageable right triangles. It’s all about seeing the hidden structures within the cube.

Conclusion

After carefully analyzing each statement, we've determined that statements 2 and 3 are correct, while statement 1 is incorrect. It's pretty neat how understanding basic geometric principles and applying them systematically can unlock the secrets of a simple cube. Remember, guys, practice makes perfect! The more you work with these concepts, the easier it will become to visualize and solve these types of problems. Keep exploring, keep questioning, and keep having fun with geometry! The key takeaways here are the accurate calculation of side length using the face diagonal, understanding the formula of space diagonal, and using the Pythagorean theorem for finding distance. Good luck, and keep on cubing!