I can help you solve the questions from the image you uploaded. Here are the eight geometry problems involving cubes, pyramids, and cuboids. Let’s go step by step to solve them.

Problem 1:

Given a cube $ABCD.EFGH$ with an unknown side length. Point $P$ is at the midpoint of side $DC$. Find the distance from point $B$ to point $P$.

Solution:

1. Assume the side length of the cube is $s$.
2. Coordinates of relevant points:
   • $B(s, 0, 0)$ (since point B is on the front bottom-left corner).
   • $D(0, s, 0)$, $C(s, s, 0)$ (since these are on the bottom face).
   • Point $P$, the midpoint of $DC$, has coordinates $\left(\frac{s}{2}, s, 0 \right)$.
3. Distance formula: = \sqrt{\left(\frac{s}{2}\right)^2 + s^2} = \sqrt{\frac{s^2}{4} + s^2} = \sqrt{\frac{5s^2}{4}} = \frac{s\sqrt{5}}{2}$$

Thus, the distance $d_{BP} = \frac{s\sqrt{5}}{2}$.

Problem 2:

You are given a square pyramid $T.ABCD$ where the height $TA = 10$ cm and the base $AB = 8$ cm. Find the distance from point $A$ to line $TD$.

Problem 3:

A square pyramid $T.ABCD$ has a slant height of 25 cm, and the length of the base is unknown. Find the distance from point $T$ to plane $ABCD$.

Problem 4:

A cube $ABCDEFGH$ has an edge length of 6 cm. If point $S$ is the intersection of diagonals $EG$ and $FH$, and point $T$ is at the midpoint of side $DH$, find the distance from point $S$ to point $T$.

Problem 5:

In a room of size $6 \, \text{m} \times 6 \, \text{m} \times 6 \, \text{m}$, a lamp is placed in the center of the ceiling. The room is modeled as a cube $ABCD.EFGH$. Determine the distance from the lamp to one of the corners on the floor.

Problem 6:

Given a cube $ABCD.EFGH$ with an edge length of $\sqrt{3}$ cm, and point $T$ on $AD$ such that $AT = 1$ cm, find the distance from point $A$ to point $B$ along line $BT$.

Problem 7:

In a cube $ABCD.EFGH$ with an edge length of 6 cm, find the distance from point $B$ to diagonal $AG$.

Problem 8:

A cuboid $PQRS.TUVW$ has dimensions $6 \, \text{cm} \times 3 \, \text{cm} \times 4 \, \text{cm}$. Point $P$ is at the midpoint of edge $VR$. Determine the distance from point $P$ to point $X$.

Which problem would you like to explore in more detail, or would you like assistance with solving all of them? Let me know!

Follow-up Questions:

1. Can you explain how to calculate the distance between two points in 3D space?
2. What is the general approach to finding the distance from a point to a line in geometry?
3. How can we determine the intersection of diagonals in a cube or pyramid?
4. How does the Pythagorean theorem apply to 3D geometry problems?
5. How do we calculate the perpendicular distance from a point to a plane?

Tip:

In geometry, visualizing the problem with a simple sketch can greatly help in understanding the relationships between points, lines, and planes.