[u]Set 4 [/u]


[b]4.1[/b] Triangle $T$ has side lengths $1$, $2$, and $\sqrt7$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral triangle.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/4a3bcf4762b97501a9575fc6972e234ffa648b.png[/img]

[b]4.2[/b]  Let $T$ be the answer from the previous problem. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$, find $k$.
[img]https://cdn.artofproblemsolving.com/attachments/a/5/168d1aa812210fd9d60a3bb4a768e8272742d7.png[/img]

[b]4.3[/b] Let $T$ be the answer from the previous problem. $T^2$ congruent squares are arranged in the configuration below (shown for $T = 3$), where the squares are tilted in alternating fashion such that they form congruent rhombuses between them. If all of the rhombuses have long diagonal twice the length of their short diagonal, compute the ratio of the total area of all of the rhombuses to the total area of all of the squares. (Hint: Rather than waiting for $T$, consider small cases and try to find a general formula in terms of $T$, such a formula does exist.)
[img]https://cdn.artofproblemsolving.com/attachments/1/d/56ef60c47592fa979bfedd782e5385e7d139eb.png[/img]

PS. You should use hide for answers.
1. **Identify the side lengths of the triangles:**
   The given triangle \( T \) has side lengths \( 1 \), \( 2 \), and \( \sqrt{7} \). We need to verify that this triangle is a right triangle. Using the Pythagorean theorem:
   \[
   1^2 + 2^2 = 1 + 4 = 5 \quad \text{and} \quad (\sqrt{7})^2 = 7
   \]
   Since \( 5 \neq 7 \), this is not a right triangle. However, we proceed with the given information.

2. **Calculate the area of the equilateral triangles:**
   - **Outer equilateral triangle:**
     The side length of the outer equilateral triangle is \( \sqrt{7} \). The area \( A \) of an equilateral triangle with side length \( a \) is given by:
     \[
     A = \frac{\sqrt{3}}{4} a^2
     \]
     Substituting \( a = \sqrt{7} \):
     \[
     A_{\text{outer}} = \frac{\sqrt{3}}{4} (\sqrt{7})^2 = \frac{\sqrt{3}}{4} \cdot 7 = \frac{7\sqrt{3}}{4}
     \]

   - **Inner equilateral triangle:**
     The side length of the inner equilateral triangle is \( 1 \). Using the same formula:
     \[
     A_{\text{inner}} = \frac{\sqrt{3}}{4} \cdot 1^2 = \frac{\sqrt{3}}{4}
     \]

3. **Compute the ratio of the areas:**
   The ratio of the area of the outer equilateral triangle to the area of the inner equilateral triangle is:
   \[
   \text{Ratio} = \frac{A_{\text{outer}}}{A_{\text{inner}}} = \frac{\frac{7\sqrt{3}}{4}}{\frac{\sqrt{3}}{4}} = \frac{7\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}} = 7
   \]

The final answer is \(\boxed{7}\)