2. The given figure is an obtuse isosceles triangle $ABC$ with base $\overline{AB}$ of length 4. The perpendicular to $\overline{AC}$ passing through point $C$ intersects segment $\overline{AB}$ at point $D$. If $|AD|=3$, determine $|CD|$.
First solution.

Sketch:

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-28.jpg?height=422&width=694&top_left_y=500&top_left_x=607)

Note. A student who draws a sketch with elements given in the problem is awarded 1 POINT out of the previous 2 POINTS. For the added height $\overline{C N}$, the remaining 1 POINT is awarded.

Since $|\Varangle D A C| = |\Varangle N A C|$ and $|\Varangle A C D| = |\Varangle C N A|$, triangles $A D C$ and $A N C$ are similar triangles (AA similarity criterion).

2 POINTS

Then their corresponding sides are proportional:

$$
\frac{|A D|}{|A C|} = \frac{|A C|}{|A N|}
$$

i.e.,

$$
\begin{aligned}
\frac{3}{a} & = \frac{a}{2} \\
a^{2} & = 6
\end{aligned}
$$

By applying the Pythagorean theorem to $\triangle A D C$, we get:

$$
\begin{gathered}
3^{2} = a^{2} + x^{2} \\
x^{2} = 9 - 6 \\
x = \sqrt{3}
\end{gathered}
$$

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