In $\Delta ABC,$ the bisector of $\angle A$ intersects $\overline{BC}$ at $D.$ If $AC = 4.2,$ $DC = 3,$ and $BC = 5,$ find $AB.$

In $\Delta PQR$,$m \angle Q : m \angle R : m \angle P = 1 : 2 : 1$. If $PQ = 2 \sqrt{6}$,then $PR = \ldots$

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $MA = 3x - 19$,$MB = x - 3$,$MC = x - 5$,and $MD = 3$,find the value of $x$.

$A$ ladder is placed leaning on a wall. Its upper end reaches to the height of $12 \, m$ on the wall and its lower end rests $9 \, m$ away from the base of the wall. Find the length of the ladder in meters.

As shown in the adjoining diagram,$\angle B$ is an obtuse angle and $\overline{AM}$ is an altitude in $\Delta ABC$. Prove that $AC^{2} = AB^{2} + BC^{2} + 2 \cdot BC \cdot BM$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $a = 9$ and $c = 12$,then $b = \dots$