In rectangle $A B C D$ with side $A D$ of length $5 \mathrm{~cm}$, point $P$ is located such that triangle $A P D$ is equilateral. Ray $A P$ intersects side $C D$ at point $E$, and segment $C E$ measures $5 \mathrm{~cm}$.

How long is segment $A E$ and what is the measure of angle $A E B$?

(L. Hozová)
Triangle $A P D$ is equilateral, so the lengths of all its sides are $5 \mathrm{~cm}$ and the measures of all its internal angles are $60^{\circ}$.

The internal angles of rectangle $A B C D$ are right angles, so the measures of angles $P D E$ and $P A B$ are $30^{\circ}$. Angle $P E D$ is the third angle of triangle $A E D$ (or also the alternate angle to angle $E A B$), so its measure is $30^{\circ}$. Angles $P D E$ and $P E D$ are congruent, thus triangle $D E P$ is isosceles and $|P D|=|P E|=5 \mathrm{~cm}$. Segment $A E$ is

$$
|A E|=|A P|+|P E|=5+5=10(\mathrm{~cm}) \text {. }
$$

Segment $C E$ is congruent to segment $C B$, so triangle $E B C$ is isosceles with a right angle at vertex $C$. The remaining two internal angles are therefore congruent with a measure of $45^{\circ}$. The measure of angle $A E B$ is

$$
|\angle A E B|=180^{\circ}-|\angle A E D|-|\angle B E C|=180^{\circ}-30^{\circ}-45^{\circ}=105^{\circ} .
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_eaf15ad62c30ffd7ab50g-3.jpg?height=449&width=1077&top_left_y=1226&top_left_x=495)

Scoring suggestion. 1 point for recognizing the isosceles nature of triangles $D P E$ and $E B C$; 1 point for calculating the length of segment $A E$ and the measure of angle $A E B$; 2 points based on the quality of the commentary.