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The problem presents a regular pyramid \( T.ABCD \) with a square base, where:

- The height of the pyramid \( T \) is \( \sqrt{3} \) cm.
- The length of the side of the square base \( AB = 6 \) cm.
- \( M \) is the intersection point of diagonals \( AC \) and \( BD \).

There are two questions:

a. The distance from point \( M \) to the line \( TD \).

b. The cosine of the angle formed by lines \( TA \) and \( TD \).

### Let's break this down:

#### Step 1: Pyramid Properties
- The square base has diagonals \( AC \) and \( BD \). Since the base is a square, the diagonals intersect at right angles at \( M \), and each diagonal has a length of \( \sqrt{2} \times \text{side of the square} = \sqrt{2} \times 6 = 6\sqrt{2} \) cm.
- \( M \) is the midpoint of both diagonals.

#### Step 2: Coordinates Setup
Assign coordinates to the vertices of the square base \( ABCD \):
- Let the base \( ABCD \) lie on the plane \( z = 0 \), centered at the origin.
- The height of the apex \( T \) is \( \sqrt{3} \) cm. Thus, \( T = (0, 0, \sqrt{3}) \).
- The vertices of the square base are \( A(-3, -3, 0) \), \( B(3, -3, 0) \), \( C(3, 3, 0) \), and \( D(-3, 3, 0) \).

#### Step 3: Calculate the Distance \( M \) to \( TD \)
- \( M \) is the midpoint of \( AC \), so \( M = (0, 0, 0) \).
- The line \( TD \) connects \( T(0, 0, \sqrt{3}) \) to \( D(-3, 3, 0) \).
- To find the distance from \( M \) to the line \( TD \), use the formula for the distance from a point to a line in 3D.

#### Step 4: Angle Between \( TA \) and \( TD \)
- To find the cosine of the angle between \( TA \) and \( TD \), use the dot product formula between the vectors \( \overrightarrow{TA} \) and \( \overrightarrow{TD} \).

I will now compute these values.Here are the results for the given problem:

a. The distance from point \( M \) to the line \( TD \) is approximately **1.60 cm**.

b. The cosine of the angle between the lines \( TA \) and \( TD \) is \( \cos(\theta) = 0.143 \).

Let me know if you need further clarification or details. Here are five related questions you can explore:

1. How is the cross product used to find the distance from a point to a line in 3D?
2. What is the geometric significance of the cosine of an angle between two vectors?
3. How do the coordinates of a pyramid affect the calculations of distances and angles?
4. How do you calculate the angle between vectors using the dot product?
5. How does the height of a pyramid affect its internal angles?

**Tip:** When calculating distances in 3D geometry, it is helpful to use vector properties like the dot and cross products for accuracy.

Diketahui limas beraturan T.ABCD dengan tinggi √3 cm, panjang AB = 6 cm, dan M adalah titik potong garis AC dan BD. Hitung jarak titik M ke garis TD dan nilai kosinus sudut yang dibentuk oleh garis TA dan TD.

This problem covers a regular pyramid T.ABCD with a base side of 6 cm and height √3 cm. Learn how to calculate the distance from point M to line TD and the cosine of the angle between lines TA and TD. Involves 3D geometry, vector cross products, and dot products.

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