Question

In the given figure, we have AB || CD || EF. If AB = 6 cm, CD = x, BD = 4 cm and DE = y cm, calculate the value of x and y.

Answer 1

Hint: As given in the question, AB is parallel to CD which is again parallel to EF. Now we have to prove triangle ADB and triangle EDF are similar. After proving this, we get the ratio of their corresponding sides and we get the value of y. Now, we take again two Triangles triangle ADE and triangle CDE. Since, both the triangles are similar, the ratio of their corresponding sides is equal. By this we get the value of x.

Complete step by step answer:

As stated in the problem, AB is parallel to CD which is again parallel to EF. So, by using the diagram we proceed for proving the triangle similar.
First, we take \[\Delta ADB\text{ and }\Delta EDF\], now we have to prove that both the triangles are similar.
In $\Delta ADB\text{ and }\Delta EDF$
$\angle ADB=\angle EDF$(Using vertical opposite angles property)
$\angle ABD=\angle FED$ (Using alternate angles property)
$\angle BAD=\angle EFD$ (Using alternate angles property)
$\therefore \Delta ADB\sim \Delta EDF$ (By using Angle Side Angle criterion)
Now, the ratio of their corresponding sides is:
\[\begin{align}
  & \Rightarrow \dfrac{BD}{DE}=\dfrac{AB}{FE} \\
 & \Rightarrow \dfrac{4}{y}=\dfrac{6}{10} \\
 & \Rightarrow 6y=40 \\
 & \Rightarrow y=\dfrac{40}{6}=\dfrac{20}{3} \\
 & \Rightarrow y=6.67cm \\
\end{align}\]
Hence, the value of y is 6.67 cm.
Similarly, $\Delta ADE\sim \Delta CDE$ (using Angle-Angle criterion)
$\begin{align}
  & \Rightarrow \dfrac{DE}{BE}=\dfrac{DC}{BA} \\
 & \Rightarrow \dfrac{6.67}{10.67}=\dfrac{x}{6} \\
 & \Rightarrow x=6\times \dfrac{6.67}{10.67} \\
 & \Rightarrow x=3.75cm \\
\end{align}$
Hence, the value of x is 3.75 cm.
Therefore, the obtained values of the x and y are 3.75 and 6.67 respectively.

Note: The key concept involved in solving this problem is the knowledge of similar triangles. Students must be careful while proving the triangle is similar using certain observable properties. The property of proportionality of sides in a similar triangle is useful in solving this problem.