Question

In the above quadrilateral ABCD,

Answer 1

Answer:

The length of segment $\overline{DO}$ is $\displaystyle \frac{24}{5}$.

Step-by-step explanation:

Start by verifying that $\triangle DAB \sim \triangle MEB$, and that $\triangle DOA \sim \triangle BOC$.

Both $\triangle DAB$ and $\triangle MEB$ contain the angle $\angle DBA$. Hence, another pair of equal angles will be sufficient to prove $\triangle DAB \sim \triangle MEB$ by the angle-angle similarity postulate.

Notice that because $\overline{AD} \; \| \; \overline{EF}$, the corresponding angles next to those two lines would be equal: $\angle MEB = \angle DAB$. Both $\angle MEB$ is in $\rm \triangle MEB$ whereas $\angle DAB$ is in $\triangle DAB$.

Hence, $\triangle DAB \sim \triangle MEB$ by the angle-angle similarity postulate.

Notice how $\angle DOA$ in $\triangle DOA$ and $\angle BOC$ in $\triangle BOC$ are two vertically opposite angles (and are thus equal to one another.) That is: $\angle DOA = \angle BOC$.

Because $\overline{AD} \| \overline{BC}$, the alternate interior angles $\angle DAO$ and $\angle BCO$ are also equal to one another. That is: $\angle DAO = \angle BCO$

Hence, $\triangle DOA \sim \triangle BOC$, also by the angle-angle similarity postulate.

Notice that $\triangle DAB \sim \triangle MEB$ implies that:$\begin{aligned} & \frac{\left| \overline{BM} \right|}{\left| \overline{BD} \right|} \\ &= \frac{\left|\overline{EB}\right|}{\left|\overline{AB}\right|} && (\triangle MEB \sim \triangle DAB) \\ &= \frac{\left|\overline{EB}\right|}{\left|\overline{AE}\right| + \left|\overline{EB}\right|} \\ &= \frac{\left|\overline{EB}\right|}{2\, \left|\overline{EB}\right| + \left|\overline{EB}\right|} && (\text{given that \left|\overline{AB}\right| = 2\,\left|\overline{EB}\right| })\\ &= \frac{1}{3}\end{aligned}$.

Therefore:

$\begin{aligned}&\left|\overline{BD}\right|\\ &= 3\, \left|\overline{BM}\right| \\ &= 12\end{aligned}$.

Similarly, $\triangle DOA \sim \triangle BOC$ implies that:

$\begin{aligned} \frac{\left| \overline{DO} \right|}{\left| \overline{BO} \right|} &= \frac{\left|\overline{AD}\right|}{\left|\overline{BC}\right|} && (\triangle DOA \sim \triangle BOC) \\ &= \frac{6}{9} = \frac{2}{3}\end{aligned}$.

Therefore, $\displaystyle \left|\overline{BO}\right|= \frac{3}{2}\, \left|\overline{DO}\right|$.

$\begin{aligned} \frac{\left| \overline{DO} \right|}{\left| \overline{BD} \right|} &= \frac{\left| \overline{DO} \right|}{\left| \overline{BO} \right| + \left| \overline{DO} \right|} \\ &= \frac{\left| \overline{DO} \right|}{(3/2)\, \left| \overline{DO} \right| + \left| \overline{DO} \right|} = \frac{2}{5}\end{aligned}$.

It was already found that $\left|\overline{BD}\right| = 12$. Therefore:

$\displaystyle \left|\overline{DO}\right| = \frac{2}{5}\, \left|\overline{BD}\right| = \frac{24}{5} = 4.8$.