The perimeter of rectangle $ABCD$ is $28$. If $AC = 10$ and $AB < BC$,find the measures of its sides.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 5.2$,$AC = 10.4$,and $BD = 3.8$,find $BC$.

$A$ $5\, m$ long ladder is placed leaning against a vertical wall such that it reaches the wall at a point $4 \,m$ high. If the foot of the ladder is moved $1.6 \,m$ towards the wall,find the distance by which the top of the ladder would slide upwards on the wall (in $m$).

The perimeter of an equilateral triangle is $18$. Then,the length of its altitude is $\ldots \ldots \ldots$

In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $m \angle B = 30^{\circ}$. Then,which of the following holds true?

In $\Delta ABC$,$\overline{AD}$ is a median. The bisectors of $\angle ADB$ and $\angle ADC$ intersect $\overline{AB}$ and $\overline{AC}$ at $E$ and $F$ respectively. Prove that $\overline{EF} \parallel \overline{BC}$.