Question

In the given figure, $AB\left\| {EF\left\| {CD.} \right.} \right.$ If $AB = 22.5cm$, $EP = 7.5cm$, $PC = 15cm$ and $DC = 27cm$. If $EF = \dfrac{m}{2},$ find $m$

Answer 1

Hint: According to given in the question we have to determine the value of $m$ when in the given figure, $AB\left\| {EF\left\| {CD.} \right.} \right.$If $AB = 22.5cm$, $EP = 7.5cm$, $PC = 15cm$ and $DC = 27cm$. If $EF = \dfrac{m}{2},$ so, first of all we have to consider the triangles DCP and PEF and in the triangles we have to apply the vertically opposite rule and alternate angle rules to make both of the triangles DCP and PEF.
Now, with the help of corresponding sides obtained we can determine the value of EF which is $EF = \dfrac{m}{2},$

Complete step-by-step answer:
Given,
$AB\left\| {EF\left\| {CD.} \right.} \right.$
$EP = 7.5cm$, $PC = 15cm$ and $DC = 27cm$.
Step 1: First of all we have to consider the triangles DCP and PEF and with the help of vertically opposite angles we can determine,
$\angle DPC = \angle EPF$ which are the vertically opposite angles
Step 2: Now, applying the alternate angle for parallel lines CD and EF.
$\angle DCP = \angle PEF$ Which are the alternate angles for parallel lines CD and EF and same as,
$\angle CDP = \angle PFE$ Which are the alternate angles for parallel lines CD and EF.
Hence, we can say that triangles DCP and PEF are congruent with the help of AAA rule.
Step 3: Now, with the help of corresponding sides,
$ \Rightarrow \dfrac{{27}}{{EF}} = \dfrac{{PC}}{{EP}}$
Step 4: On substituting all the values in the expression as obtained in the solution step 3.

$
   \Rightarrow \dfrac{{27}}{{EF}} = \dfrac{{15}}{{7.5}} \\
   \Rightarrow EF = \dfrac{{27 \times 7.5}}{{15}} \\
   \Rightarrow EF = 13.5cm
 $
Step 5: Now, on substituting the value of EF in $EF = \dfrac{m}{2},$
$
   \Rightarrow 13.5 = \dfrac{m}{2} \\
   \Rightarrow m = 13.5 \times 2 \\
   \Rightarrow m = 27cm
 $

Hence, with the help of AAA rule we have obtained the value $m = 27cm$.

Note: If all the three angles are equal to all the three angles of another triangle then we can say that the both of the triangles are congruent.
If all the three sides are equal to all the three sides of another triangle then we can say that the both of the triangles are congruent.
If all the two angles and one side of a triangle are equal to all the two angles and one side of another triangle then we can say that the both of the triangles are congruent.