## Task B-3.7.

Let $A B C D$ be a convex quadrilateral such that $|A B|=2 \sqrt{3}, |B C|=3, |C D|=1$, $\angle D A B=90^{\circ}$ and $\angle A B C=30^{\circ}$. Determine $\angle B C D, \angle C D A$ and $|A D|$.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_d4daff78382705a7c34eg-21.jpg?height=374&width=733&top_left_y=1618&top_left_x=593)

Consider triangle $ABC$. By applying the Law of Cosines, we get

$|AC|^2 = |AB|^2 + |BC|^2 - 2 \cdot |AB| \cdot |BC| \cos \angle ABC = 12 + 9 - 2 \cdot 2 \sqrt{3} \cdot 3 \cdot \cos 30^\circ = 3$.

Thus, $|AC| = \sqrt{3}$.

2 points

By applying the Law of Sines, we calculate $\angle CAB$.

$\frac{|BC|}{\sin \angle CAB} = \frac{|AC|}{\sin \angle ABC}$, so $\sin \angle CAB = \frac{3 \cdot \frac{1}{2}}{\sqrt{3}} = \frac{\sqrt{3}}{2}$.

1 point

It follows that $\angle CAB = 60^\circ$ or $120^\circ$.

1 point

The case $\angle CAB = 120^\circ$ is not possible because the quadrilateral $ABCD$ is convex, so $\angle CAB < \angle DAB = 90^\circ$. Therefore, $\angle CAB = 60^\circ$.

Furthermore, $\angle DAC = 90^\circ - \angle CAB = 30^\circ$.

Notice that $\angle BCA = 180^\circ - 30^\circ - 60^\circ = 90^\circ$.

Consider triangle $ACD$. By applying the Law of Sines, we get

$\frac{|AC|}{\sin \angle CDA} = \frac{|CD|}{\sin \angle DAC}$, so $\sin \angle CDA = \frac{\sqrt{3} \cdot \frac{1}{2}}{1} = \frac{\sqrt{3}}{2}$

It follows that $\angle CDA = 60^\circ$ or $\angle CDA = 120^\circ$.

1 point

The first case is not possible because then $\angle BCD = 360^\circ - 60^\circ - 90^\circ - 30^\circ = 180^\circ$. Therefore, $\angle CDA = 120^\circ$.

Furthermore, $\angle ACD = 180^\circ - \angle DAC = 30^\circ$, so triangle $ACD$ is isosceles, meaning $|AD| = |CD| = 1$.

It is also necessary to calculate the measure of angle $\angle BCD$.

$\angle BCD = \angle BCA + \angle ACD = 90^\circ + 30^\circ = 120^\circ$.

1 point

## ZUPANIJSKO NATJECANJE IZ MATEMATIKE

## 4th grade - high school - B variant

March 29, 2021.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE TASK, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH APPROPRIATELY.