Question

In the figure given, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O then,

  \[\begin{align}
  & \left( A \right)\dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{BO}{CO} \\
 & \left( B \right)\dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{AO}{BO} \\
 & \left( C \right)\dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{AO}{DO} \\
 & \left( D \right)\dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{CO}{DO} \\
\end{align}\]

Answer 1

Hint: In order to find the ratio, we will first find the area of triangle \[ABC\] and then that of the area \[DBC\] , and then the ratio, \[\dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}\] . Now as the two triangles have the same base \[BC\] , therefore, it is easier to calculate the ratio.
Formula used: The formula used is that for the area of the triangle.
i.e. \[\begin{align}
  & A=\dfrac{1}{2}\times base\times height \\
 & A=\dfrac{1}{2}\times b\times h \\
\end{align}\]

Complete step-by-step answer:
Draw \[AP\] Perpendicular to \[BC\] and \[DQ\] perpendicular to \[BC\]

 The area of triangle is given by the formula
 \[\begin{align}
  & A=\dfrac{1}{2}\times base\times height \\
 & A=\dfrac{1}{2}\times b\times h \\
\end{align}\]
In triangle \[ABC\] ,
The base is:
 \[BC=b1\]
And height is :
 \[h1=AP\]
Therefore, the area of triangle \[ABC\] is
 \[ar\left( ABC \right)=\dfrac{1}{2}\times AP\times BC\]
Now, in triangle, \[DBC\]
The base is :
 \[b2=BC\]
And the height is :
 \[h2=DQ\]
So, the area of the triangle \[DBC\] is
 \[ar\left( DBC \right)=\dfrac{1}{2}\times DQ\times BC\]
Now, calculating the ratio,
 \[\begin{align}
  & \dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{\dfrac{1}{2}\times AP\times BC}{\dfrac{1}{2}\times DQ\times BC} \\
 & \Rightarrow \dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{AP}{DQ} \\
\end{align}\]
Therefore, the ratio of the area of the two triangles \[ABC\] and \[DBC\] is \[\dfrac{AP}{DQ}\]
Now, taking triangles \[APO\] and \[DQO\] .
 \[\begin{align}
  & \left| \!{\underline {\,
  APO \,}} \right. =\left| \!{\underline {\,
  DQO \,}} \right. =90 \\
 & \left| \!{\underline {\,
  AOP \,}} \right. =\left| \!{\underline {\,
  DOQ \,}} \right. =V.O.A \\
\end{align}\]
Therefore, \[APO\] is congruent to \[DQO\]
 \[\begin{align}
  & \Rightarrow \dfrac{AP}{DQ}=\dfrac{AO}{DO} \\
 & \Rightarrow \dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{AP}{DQ}=\dfrac{AO}{DO} \\
\end{align}\]
From the options given in the question, the third option gives this ratio
i.e. \[\left( C \right)\dfrac{ar\left( ABC \right)}{ar\left( DBC \right)}=\dfrac{AO}{DO}\]
So, the option \[\left( C \right)\] is correct.
So, the correct answer is “Option C”.

Note: The height of the triangle is the perpendicular drawn from a vertex to the base of the triangle. While the base of the triangles in this trapezoid are the diagonals.
  \[\dfrac{AP}{DQ}=\dfrac{AO}{DO}\] .
This step is the most important one. This is the property of congruence of two triangles, that is if two triangles are congruent, the ratio of the corresponding sides are equal. This property has been used to reach the final result, without it, we got the ratio that doesn’t match a single option.