B1. Let $A B C D$ be a trapezoid with bases $A B$ and $C D$. Let $|A B|=6.5 \, \text{cm}, |B C|=4.3 \, \text{cm}$, $|A D|=3.8 \, \text{cm}$, and $|A C|=5.3 \, \text{cm}$. Calculate the size of the angle $\angle C B A$, the height of the trapezoid $A B C D$, and its area.
B1. Let $a=|A B|, b=|B C|, c=|C D|, d=|A D|, e=|A C|$. Let $X$ and $Y$ be the feet of the altitudes from $C$ and $D$ respectively, and let $x=|X B|$ and $y=|A Y|$. Using the cosine rule, we calculate the angle $\cos \beta=\frac{a^{2}+b^{2}-e^{2}}{2 a b} \Rightarrow \beta \doteq 54.26^{\circ} \doteq 54^{\circ} 16'$. In the right triangle with sides $|X C|,|X B|$, and $|B C|$ and angle $\beta$, we calculate the height of the trapezoid $v=b \cdot \sin \beta \doteq 3.5 \mathrm{~cm}$ and the side $x=\sqrt{b^{2}-v^{2}} \doteq 2.5 \mathrm{~cm}$. In the other right triangle with sides $|A Y|,|Y D|$, and $|A D|$, using the Pythagorean theorem, we calculate the side $y=\sqrt{d^{2}-v^{2}} \doteq 1.5 \mathrm{~cm}$ and thus obtain the length of the side $c=a-x-y \doteq 2.5 \mathrm{~cm}$. Using the formula $S=\frac{a+c}{2} \cdot v$, we calculate the area of the trapezoid $S \doteq 15.75 \mathrm{~cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-24.jpg?height=55&width=1739&top_left_y=1503&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-24.jpg?height=48&width=1739&top_left_y=1552&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-24.jpg?height=51&width=1739&top_left_y=1599&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-24.jpg?height=57&width=1742&top_left_y=1645&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-24.jpg?height=66&width=1736&top_left_y=1692&top_left_x=177)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-24.jpg?height=52&width=1736&top_left_y=1753&top_left_x=177)
The formula for the area $S=\frac{a+c}{2} \cdot v \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots . \ldots$ point The calculated area of the trapezoid $S \doteq 15.75 \mathrm{~cm}^{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$