Cube Geometry: Solving Area & Parallelism Problems

Hey guys! Let's dive into a classic geometry problem involving a cube. We'll be working with a cube labeled ABCDA'B'C'D', and we've got some specific information to help us solve the problems. Buckle up, because we're going to use our knowledge of 3D geometry to calculate areas and prove some cool relationships. This is all about applying what we know and visualizing the cube in our heads. Ready to get started? Let's go!

Understanding the Cube and Given Information

Okay, so the problem starts by introducing us to a cube, which is super important. We need to know that all the sides are equal and all the angles are right angles. We are also told that B'C' = 10√2 cm. This is a face diagonal of the cube. Remember that the cube has all edges that are the same length. The face diagonal is the line segment that connects two opposite corners on the same face of the cube. From the formula for the face diagonal, we can find out the length of the edges of the cube. We're going to use this value to find the area of a triangle and prove that two planes are parallel. We are asked to find the area of triangle BDC' and to show that plane AB'D' is parallel to plane BDC'. This problem is a great way to improve our 3D geometry skills. We'll break down the question step by step, so even if you're not a math whiz, you should be able to follow along. Let’s start by figuring out what the side length of the cube is. Remember, the face diagonal, such as B'C', creates a right-angled triangle with two edges of the cube. By using the Pythagorean theorem, we can relate the face diagonal to the edge length. Then, once we have the side length, we can go on and solve the other parts of the question. Now, let’s get into the actual calculations!

Finding the Side Length of the Cube

Alright, let's find the side length of the cube, which we'll call 'a'. We know that B'C' = 10√2 cm. The diagonal of a square (which is what each face of the cube is) can be calculated using the formula: diagonal = a√2, where 'a' is the side length. We know the diagonal (B'C') is 10√2 cm, so we can set up the equation a√2 = 10√2. To find 'a', we divide both sides by √2. So, a = 10 cm. Therefore, each edge of the cube is 10 cm long. This is crucial information, because now we can use this information to tackle the first part of our problem: finding the area of triangle BDC'. This is the foundation upon which we’ll build our solution. With the side length in hand, we have everything we need to move forward and solve the remaining parts of this problem. Remember that in geometry, details matter. Let’s keep this in mind as we find the area of triangle BDC'. We are going to visualize it from many different perspectives. Let’s get to the next step.

Calculating the Area of Triangle BDC'

Now, let's calculate the area of triangle BDC'. This might seem a bit tricky at first, but we can do it! Triangle BDC' is a triangle within the cube. Imagine cutting across the cube to form this triangle. To find the area, we're going to use the formula for the area of a triangle: Area = (1/2) * base * height. We can consider BD as the base of the triangle. BD is a face diagonal, and we already know that the edge of the cube is 10 cm. The face diagonal of the cube is given by the formula: diagonal = a√2. So, BD = 10√2 cm. Next, we need to find the height. The height of the triangle is the perpendicular distance from the point C' to the line BD. If we analyze the cube, we can see that the line segment from C' to the midpoint of BD is perpendicular to BD and is also an edge of the cube. Thus, the height of our triangle is CC', which is the edge of the cube (10 cm). However, to be more precise, we have to recognize that the height is the perpendicular distance from C' to the line BD. Since we know that BD is a face diagonal, we can use the formula for the area of an equilateral triangle. We can view the triangle BDC' as an equilateral triangle (all sides are equal in length, each being a face diagonal). We know that the length of the face diagonal is 10√2 cm. The formula for the area of an equilateral triangle is (√3/4) * side^2. Thus, the area will be: Area = (√3/4) * (10√2)^2 = (√3/4) * 200 = 50√3 cm². So, the area of triangle BDC' is 50√3 cm². We've successfully calculated the area! Now, let’s go and prove that the planes AB'D' and BDC' are parallel.

Proving Parallelism: (AB'D') || (BDC')

Alright, let’s get to the fun part and demonstrate that plane AB'D' is parallel to plane BDC'. To prove that two planes are parallel, we can show that two intersecting lines in one plane are parallel to the other plane. This is like proving that two walls in a room never intersect. Look at the lines in the cube! First of all, we know that the line segment BD is in plane BDC', and the line segment B'D' is in the plane AB'D'. Since B'D' and BD are face diagonals of parallel faces of the cube, they are parallel. Because they are face diagonals of the opposite faces of the cube, we can tell that B'D' || BD. Now, let’s consider the line segment AB' and the line segment C'D. The line segment AB' is in the plane AB'D', and the line segment C'D is in the plane BDC'. These lines are not only in different planes but also parallel to each other. AB' and C'D are diagonals in parallel rectangles of the cube. They are also equal in length. Therefore, AB' || C'D. Since AB' is parallel to C'D and B'D' is parallel to BD, and since the lines intersect, we can conclude that the planes AB'D' and BDC' are parallel. This completes our proof! We have demonstrated that the two planes are parallel. We've used properties of parallel lines and planes to solve the problem and shown that two planes are parallel. We have now finished the problem! Congratulations!

Conclusion: Recap and Key Takeaways

So, we did it, guys! We successfully tackled the problem of finding the area of the triangle and proving the parallelism. We started by figuring out the side length of the cube, then we calculated the area of the triangle BDC', and finally, we demonstrated that the two planes are parallel. The main thing we used was our understanding of the cube's properties, like the relationship between edge length and face diagonals, and the properties of parallel lines and planes. We used the formula for the area of a triangle. We used the properties of parallel lines and the concepts of 3D geometry to solve the problem. Key takeaways: Understand how to find the area of a triangle, especially in 3D geometry; grasp the concept of parallelism in 3D; and know the relationships between edges, face diagonals, and space diagonals in a cube. Keep practicing, and you'll become a 3D geometry expert in no time! Keep up the good work and keep learning! We'll see you next time!