Question

In figure (1) and (2), \[DE||BC\]. Find EC in (1) and AD in (2).

Answer 1

Hint: For the above question, we will use Thales theorem, which states that a line is drawn parallel to one side of a triangle, intersecting the other two sides in distinct parts, then the other two sides are divided in the same ratio.

Complete step-by-step answer:
We have been given the figure (1) as follows:

As we know that according to Thales theorem if a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, then the other two sides are divided in the same ratio.
\[\begin{align}
  & \dfrac{AD}{BD}=\dfrac{AE}{EC} \\
 & \dfrac{1.5}{3}=\dfrac{1}{EC} \\
\end{align}\]
Hence, on cross multiplication, we will get the equation as follows:
\[\begin{align}
  & 1.5(EC)=3 \\
 & EC=2cm \\
\end{align}\]
We also have been given figure (2) as follows:

According to Thales theorem, we already know that,
\[\dfrac{AD}{BD}=\dfrac{AE}{EC}\]
Hence, on substituting the value of BD, AE and EC in the above expression, we will get the equation as follows:
\[\begin{align}
  & \dfrac{AD}{7.2}=\dfrac{1.8}{5.4} \\
 & \dfrac{AD}{7.2}=\dfrac{1}{3} \\
\end{align}\]
Then, on cross multiplication, we will get the equation as follows:
\[\begin{align}
  & 3(AD)=7.2 \\
 & AD=\dfrac{7.2}{3} \\
 & AD=2.4cm \\
\end{align}\]
Therefore, the value of EC in figure (1) is equal to 2 cm and AD in figure (2) is equal to 2.4 cm.

Note: Thales theorem is also known as The Basic Proportionality Theorem. We have to be extremely careful while doing the cross multiplication of the terms in the above mentioned equations, as there is a slight chance that we might make a mistake here.