Let us break the problem down and solve it step by step.

Given:

• Rectangle $ABCD$ with: $AB = 6, \quad AD = 4.$
• The goal is to:
  1. Determine $AC$ and $AD$.
  2. Deduce the approximate value of the angle $\widehat{CAD}$ to the nearest tenth of a degree.

Step 1: Calculate $AC$ (diagonal of the rectangle)

The diagonal $AC$ of rectangle $ABCD$ can be calculated using the Pythagorean theorem: $AC = \sqrt{AB^2 + AD^2}.$

Substitute the given values: $AC = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52}.$

Simplify: $AC = \sqrt{52} \approx 7.21.$

Step 2: Angle $\widehat{CAD}$

The angle $\widehat{CAD}$ can be calculated using the tangent ratio: $\tan(\widehat{CAD}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AD}{AC}.$

Substitute the known values: $\tan(\widehat{CAD}) = \frac{4}{6} = 0.6667.$

Now, find $\widehat{CAD}$ by taking the arctangent: $\widehat{CAD} = \arctan(0.6667).$

Using a calculator: $\widehat{CAD} \approx 33.7^\circ.$

Final Answer:

1. $AC \approx 7.21$.
2. $\widehat{CAD} \approx 33.7^\circ$.

Would you like further explanations or details? Here are related questions to expand your understanding:

1. What is the formula for the diagonal of a rectangle in terms of its sides?
2. How does the Pythagorean theorem relate to the geometry of rectangles?
3. How can the tangent ratio be applied in other right triangles?
4. What are alternative ways to compute angles in triangles?
5. How does rounding impact the precision of geometric calculations?

Tip: Remember, the tangent function is particularly useful for finding angles in right triangles when you know the lengths of two sides.