https://r2.erweima.ai/mathbot/image/1726281186230y831bpjx.jpg

Let's break down and solve the problem from the image.

The vertices of the pyramid \( E - ABCD \) are given as:
- \( A(2, -2, 4) \)
- \( B(2, 4, 4) \)
- \( C(-4, 4, 4) \)
- \( D(-4, -2, 4) \)
- \( E(-1, 1, 8) \)

### a) The sum of the distances from point \( A(2, -2, 4) \) to each of the coordinate planes

The distance from a point \( (x, y, z) \) to a coordinate plane is:
- Distance to the \( yz \)-plane (where \( x = 0 \)): \( |x| \)
- Distance to the \( xz \)-plane (where \( y = 0 \)): \( |y| \)
- Distance to the \( xy \)-plane (where \( z = 0 \)): \( |z| \)

For point \( A(2, -2, 4) \):
- Distance to the \( yz \)-plane = \( |2| = 2 \)
- Distance to the \( xz \)-plane = \( |-2| = 2 \)
- Distance to the \( xy \)-plane = \( |4| = 4 \)

Thus, the sum of the distances is:
\[
2 + 2 + 4 = 8
\]

### b) The area of triangle \( ABC \)

The coordinates of the vertices of triangle \( ABC \) are:
- \( A(2, -2, 4) \)
- \( B(2, 4, 4) \)
- \( C(-4, 4, 4) \)

We can use the formula for the area of a triangle given its vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) in 3D space:

\[
\text{Area} = \frac{1}{2} \| \vec{AB} \times \vec{AC} \|
\]

First, we calculate the vectors:
- \( \vec{AB} = B - A = (2 - 2, 4 - (-2), 4 - 4) = (0, 6, 0) \)
- \( \vec{AC} = C - A = (-4 - 2, 4 - (-2), 4 - 4) = (-6, 6, 0) \)

Next, we calculate the cross product \( \vec{AB} \times \vec{AC} \):
\[
\vec{AB} \times \vec{AC} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
0 & 6 & 0 \\
-6 & 6 & 0
\end{vmatrix}
= \hat{i}(6 \times 0 - 0 \times 6) - \hat{j}(0 \times 0 - (-6) \times 0) + \hat{k}(0 \times 6 - (-6) \times 6)
\]
\[
\vec{AB} \times \vec{AC} = \hat{i}(0) - \hat{j}(0) + \hat{k}(36) = (0, 0, 36)
\]

The magnitude of the cross product is:
\[
\| \vec{AB} \times \vec{AC} \| = \sqrt{0^2 + 0^2 + 36^2} = 36
\]

Thus, the area of triangle \( ABC \) is:
\[
\text{Area} = \frac{1}{2} \times 36 = 18
\]

### c) The volume of the pyramid

The formula for the volume of a pyramid with a triangular base is:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base of the pyramid is the quadrilateral \( ABCD \), but for simplicity, we can treat one of the triangular sides like \( ABC \) as the base.

From part (b), the area of the triangular base \( ABC \) is \( 18 \).

The height is the perpendicular distance from the apex \( E(-1, 1, 8) \) to the plane containing \( ABC \). The equation of the plane through \( A(2, -2, 4) \), \( B(2, 4, 4) \), and \( C(-4, 4, 4) \) can be found since all points have \( z = 4 \), implying the plane is parallel to the \( xy \)-plane and its equation is:
\[
z = 4
\]

The height of the pyramid is the distance from \( E(-1, 1, 8) \) to the plane \( z = 4 \), which is simply:
\[
|8 - 4| = 4
\]

Thus, the volume of the pyramid is:
\[
V = \frac{1}{3} \times 18 \times 4 = 24
\]

### Summary of answers:
- a) The sum of the distances from \( A \) to the coordinate planes is **8**.
- b) The area of triangle \( ABC \) is **18**.
- c) The volume of the pyramid is **24**.

Would you like more details or have any further questions?

Here are 5 related questions you might explore:
1. How would you calculate the centroid of the pyramid?
2. How would the volume change if the apex \( E \) were moved?
3. What is the distance between vertex \( E \) and vertex \( A \)?
4. How can we determine the equation of the plane through points \( A \), \( B \), and \( C \)?
5. Can you generalize the formula for the volume of a pyramid with a polygonal base?

**Tip:** When calculating areas or volumes in 3D, using vector cross products and plane equations can simplify the geometry.

Given the vertices of the pyramid E - ABCD, where A(2, -2, 4), B(2, 4, 4), C(-4, 4, 4), D(-4, -2, 4), and E(-1, 1, 8), determine: (a) the sum of the distances from point A to each of the coordinate planes, (b) the area of triangle ABC, and (c) the volume of the pyramid.

Learn how to find the sum of the distances from a point to coordinate planes, calculate the area of a triangle in 3D, and determine the volume of a pyramid using vector cross products. This geometry problem is designed for Grades 10-12 students.

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