Question

please solve the problem ​

Answer 1

Given :

• $\sf{\dfrac{AO}{OC}\:=\:\dfrac{1}{2}}$

• $\sf{\dfrac{BO}{OD}\:=\:\dfrac{1}{2}}$
• AB = 5 cm.

To Find :

• Length of DC.

Solution :

$\sf{\dfrac{AO}{OC}\:=\:\dfrac{1}{2}}$

$\sf{2AO\:=\:OC\:\:\:(1)}$

Now, in Δ AOB and ΔDOC,

$\sf{\angle\:AOB\:=\:\angle\:DOC}$

$\bold{\big[Vertically\:Opposite\:Angles\big]}$

$\sf{\angle\:OAB\:=\:\angle\:OCD}$

$\bold{\big[Alternate\:Angles\big]}$

•°•ΔOAB ~ Δ OCD

$\bold{\big[By\:AA\:test\:of\:similarlity\big]}$

•°• $\sf{\dfrac{AO}{OC}\:=\:\dfrac{AB}{CD}\:=\:\dfrac{BO}{DO}}$

$\bold{\big[Corresponding\:sides\:of\:similar\:triangle\:are\:in\:proportion\big]}$

$\sf{\dfrac{AO}{OC}\:=\:\dfrac{AB}{CD}}$

$\sf{\dfrac{AO}{2AO}\:=\:\dfrac{5}{DC}}$

$\bold{\big[From\:equation\:(2)\big]}$

$\sf{\dfrac{1}{2}\:=\:\dfrac{5}{DC}}$

$\sf{DC\:=\:5\:\times\:2}$

$\sf{DC\:=\:10\:cm}$

$\large{\boxed{\bold{Length\:of\:DC\:=\:10\:cm}}}$