Question

In fig , $ \left( i \right) $ and $ \left( {ii} \right) $ , $ DE\parallel BC $ . Find $ EC $ in $ \left( i \right) $ and $ AD $ in $ \left( {ii} \right) $ .

Answer 1

Hint: Use the property that if the line $ DE $ is parallel to the base of the triangle then it divides the sides of the triangle in the same ratios. Make the ratios equal and find the length of the required side of the triangle.

Complete step-by-step answer:
I.Firstly consider the triangle $ \left( i \right) $ in the figure. As the side $ DE $ is parallel to the base of the triangle $ BC $ .
So, the side $ DE $ divides the sides of the triangle in the same ratio.
So, consider the ratio of the sides:
 $
  \dfrac{{AD}}{{DB}} = \dfrac{{AE}}{{EC}} \\
  EC = AE \times \dfrac{{DB}}{{AD}} \;
  $
Substitute the lengths of the sides $ AE $ , $ DB $ and $ AD $ as given in the diagram.
 $
\Rightarrow EC = AE \times \dfrac{{DB}}{{AD}} \\
   = 1\;cm \times \dfrac{{3\;cm}}{{1.5\;cm}} \\
   = 2\;cm \;
  $
So, the length of the side $ EC $ in the triangle $ \left( i \right) $ given in the figure is equal to $ 2\;cm $.
So, the correct answer is “$ 2\;cm $”.

II.Firstly consider the triangle $ \left( {ii} \right) $ in the figure. As the side $ DE $ is parallel to the base of the triangle $ BC $ .
So, the side $ DE $ divides the sides of the triangle in the same ratio.
So, consider the ratio of the sides:
 $
  \dfrac{{AD}}{{DB}} = \dfrac{{AE}}{{EC}} \\
  AD = DB \times \dfrac{{AE}}{{EC}} \;
  $
Substitute the lengths of the sides $ AE $ , $ DB $ and $ AD $ as given in the diagram.
 $
\Rightarrow AD = DB \times \dfrac{{AE}}{{EC}} \\
   = 7.2\;cm \times \dfrac{{1.8\;cm}}{{5.4\;cm}} \\
   = 7.2\;cm \times \dfrac{1}{{3\;cm}} \\
   = 2.4\;cm \;
  $
So, the length of the side $ AD $ in the triangle $ \left( i \right) $ given in the figure is equal to $ 2.4\;cm $ .
So, the length of the side $ EC $ in the triangle $ \left( i \right) $ given in the figure is equal to $ 2\;cm $ and the length of the side $ AD $ in the triangle $ \left( i \right) $ given in the figure is equal to $ 2.4\;cm $ .
So, the correct answer is “$ 2.4\;cm $”.

Note: The property that the side parallel to its base divides the sides of the triangle in the same ratio is a property of triangles. Remember to check the units of the length of each side as it's mandatory and can create a problem.