Question

In the figure $AD\parallel BC$, find the value of x

Answer 1

Step 1: Check the similarity of triangles $∆AOD$ and $∆BOC$

Given $AD$ is parallel to $BC$$\Rightarrow AD\parallel BC$
The length of $AO=3$
The length of $BO=3x-19$
The length of $DO=x-5$
The length of $OC=x-3$

AAA similarity: If in two triangles, the corresponding angles are equal, then the triangles are similar.

In $∆AOD,∆BOC$

$\angle DAO=\angle OCB$ [Alternate angles]

$\angle ADO=\angle CBO$ [Alternate angles]

$\angle AOD=\angle BOC$ [Vertical opposite angle]

$∆AOD~∆BOC$ [by AAA similarity]

Step 2: Calculate the value of $x$

Similar Triangles property: If two triangles are similar, then their corresponding sides are proportional.

Since $∆AOD~∆BOC$, thus

$\frac{AO}{OC}=\frac{OD}{OB}$

$\Rightarrow \frac{3}{x-3}=\frac{x-5}{3x-19}$

$\Rightarrow 3(3x-19)=(x-3)(x-5)$

$\Rightarrow 9x-57=x^2-8x+15$

$\Rightarrow (x-9)(x-8)=0$

$\Rightarrow x=8,9$

Hence, the value of $x$ is $8$ or $9$.