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We are dealing with a cube where all edges are 1 unit long. The points of interest are \( C \), \( A \), and \( E \), and we are asked to solve two parts of the problem:

### a) What is the measure of angle \( \angle CAE \)?
- From the diagram, \( C \), \( A \), and \( E \) appear to form a triangle on the cube. The points are located on different faces of the cube.
  
### b) What is the perimeter of triangle \( CAE \)?
- The perimeter of triangle \( CAE \) is simply the sum of the lengths of the sides \( CA \), \( AE \), and \( CE \).

#### Step 1: Determine the coordinates of the points in 3D space
Since the cube has unit edges, we can assign coordinates to each of the vertices:

- \( A = (0, 0, 0) \)
- \( C = (1, 1, 0) \)
- \( E = (0, 1, 1) \)

#### Step 2: Measure of \( \angle CAE \)
We can use the dot product formula to find the measure of angle \( CAE \). The vectors we need are:
- \( \overrightarrow{CA} = A - C = (0, 0, 0) - (1, 1, 0) = (-1, -1, 0) \)
- \( \overrightarrow{EA} = A - E = (0, 0, 0) - (0, 1, 1) = (0, -1, -1) \)

The formula for the angle between two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is:
\[
\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}
\]

I'll calculate the angle next.The measure of \( \angle CAE \) is approximately \( 60^\circ \).

#### Step 3: Perimeter of triangle \( CAE \)
Now, we calculate the lengths of the sides \( CA \), \( AE \), and \( CE \).

- The length of \( CA \) is the distance between points \( C(1,1,0) \) and \( A(0,0,0) \).
- The length of \( AE \) is the distance between points \( A(0,0,0) \) and \( E(0,1,1) \).
- The length of \( CE \) is the distance between points \( C(1,1,0) \) and \( E(0,1,1) \).

I will now calculate these distances.The perimeter of triangle \( CAE \) is approximately \( 4.24 \) units. Each side of the triangle, \( CA \), \( AE \), and \( CE \), has a length of about \( 1.41 \) units.

### Summary:
- a) The measure of \( \angle CAE \) is \( 60^\circ \).
- b) The perimeter of triangle \( CAE \) is approximately \( 4.24 \) units.

Would you like more details on any step or additional clarification?

Here are 5 related questions to consider:
1. How is the dot product formula used to find angles between vectors?
2. How can the Pythagorean theorem be applied in 3D space?
3. What are some properties of a regular tetrahedron related to this cube?
4. How can you calculate the area of triangle \( CAE \) in 3D space?
5. What other geometric shapes can you form with points on a cube?

**Tip:** When calculating angles in 3D geometry, always check that the vectors are in the correct coordinate system before applying the dot product formula.

In this cube with 1 unit edges:
a) What is the measure of angle CAE?
b) What is the perimeter of triangle CAE?

Solve for the measure of angle CAE and the perimeter of triangle CAE in a cube with 1 unit edges. Explore key geometric concepts like vector dot products and distance formulas in 3D space. Suitable for high school geometry students.

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