A "knobbed" deltoid has sides of $3 \mathrm{~cm}$ and $4 \mathrm{~cm}$. Its angles are variable. What is the length of the diagonals of the convex deltoid if its area is half of the maximum possible value?
Solution. Let $A B=B C=3 \mathrm{~cm}, C D=D A=4 \mathrm{~cm}$. The area of triangle $A B D$,

$$
t=\frac{3 \cdot 4 \cdot \sin \alpha}{2}
$$

is half the area of the kite, so the area of the kite is: $12 \cdot \sin \alpha$. This is maximized when $\sin \alpha=1$, hence $\alpha=90^{\circ}$ and $T=12 \mathrm{~cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_22fd0c882b9a24349637g-1.jpg?height=388&width=326&top_left_y=440&top_left_x=886)

Figure 1

The maximum achievable area value is half

$$
6=3 \cdot 4 \cdot \sin \alpha
$$

from which $\sin \alpha=\frac{1}{2}$. The two values for $\alpha$ are $\alpha_{1}=30^{\circ}, \alpha_{2}=150^{\circ}$.

We know that the height of an equilateral triangle with a side length of $4 \mathrm{~cm}$ is $2 \sqrt{3}>3$, so in the triangle with two sides of $4 \mathrm{~cm}$ and $3 \mathrm{~cm}$ and an included angle of $30^{\circ}$, the angle opposite the $4 \mathrm{~cm}$ side is obtuse. This means that when $\alpha=30^{\circ}$, the kite is concave, while when $\alpha=150^{\circ}$, it is convex (Figure 2). According to the problem statement, we need to determine the diagonals of the latter quadrilateral.
![](https://cdn.mathpix.com/cropped/2024_05_02_22fd0c882b9a24349637g-1.jpg?height=598&width=582&top_left_y=1322&top_left_x=760)
Figure 2

Let's calculate the lengths of the diagonals using the cosine rule:

$$
B D^{2}=3^{2}+4^{2}-2 \cdot 3 \cdot 4 \cdot \cos 150^{\circ}
$$

from which

$$
B D=\sqrt{25+12 \sqrt{3}} \approx 6.766
$$

The area of the kite is $t=\frac{A C \cdot B D}{2}$. Substitute the values of $t$ and $B D$ and solve for $A C$:

$$
A C=\frac{2 t}{B D}=\frac{2 \cdot 6}{\sqrt{25+12 \sqrt{3}}}=\frac{12 \cdot \sqrt{193} \cdot \sqrt{25-12 \sqrt{3}}}{193} \approx 1.773
$$

Note. Some solvers overlooked that the conditions are satisfied by two kites, but the problem statement only refers to the convex kite.