Question

In the given figure $DA\perp AB,CB\perp AB$ and $OM\perp AB$. If AO = 5.4 cm, OC = 7.2 cm and BO = 6 cm, then the length of DO is:

• A. 4.5 cm
• B. 4 cm
• C. 5 cm
• D. 6.5 cm

Answer 1

Answer: (A)

4.5 cm

Consider $\Delta CABand\Delta OAM$.
$\angle OMA=\angle CBA=\angle {90}^0$
$\angle OAM=\angle CAB\left[commonangle\right]$
$\angle AOM=\angle ACB\left[correspondingangles\right]$ [two lines perpendicular to the base are parallel to one another]
Hence $\Delta OAM\sim \Delta CAM$ by AAA.
So, $\frac{AO}{AC}=\frac{AM}{AB}=\frac{5.4}{12.6}$
Similarly, consider $\Delta DBAand\Delta OBM$.
$\angle OMB=\angle DAB=\angle {90}^0$
$\angle OBM=\angle DBA\left[commonangle\right]$
$\angle BOM=\angle BDA\left[correspondingangles\right]$ [two lines perpendicular to the base are parallel to one another]
Hence $\Delta OBM\sim \Delta DBA$ by AAA.
So, $\frac{AO}{AC}=\frac{AM}{AB}=\frac{5.4}{12.6}and\frac{BO}{BD}=\frac{BM}{AB}=\frac{AB-AM}{AB}=1-\frac{AM}{AB}=1-\frac{\mathrm{5.4.}}{12.6}\frac{BO}{BD}=\frac{7.2}{12.6}⟹\frac{6}{BD}=\frac{7.2}{12.6}⟹BD=10.5IfBD=10.5andOB=6thenDO=4.5$.