Question

In the figure, \[AE:EB = 1:2\] and \[BD:DC = 5:3\]. Find \[EG:GC\].
A) \[1:3\]
B) \[5:3\]
C) \[5:9\]
D) \[1:9\]

Answer 1

Hint:
We will first make an appropriate construction within the triangle using the given information. Then, we will apply the Thales theorem to find the required ratio. Thales theorem states that “If a straight line is drawn parallel to a side of a triangle, then it divides the other two sides proportionately”.

Complete step by step solution:
We are given that \[AE:EB = 1:2\] and \[BD:DC = 5:3\]. We are supposed to find \[EG:GC\].
Let us first make a construction within $\vartriangle ABD$. We will draw \[EP\parallel AD\]. We will make use of Thales theorem to approach this problem.

Let us apply Thales theorem to $\vartriangle ABD$. Here, \[EP\parallel AD\]. So, by Thales theorem, we have
\[\dfrac{{BE}}{{EA}} = \dfrac{{BP}}{{PD}}\]
\[ \Rightarrow \dfrac{{BP}}{{PD}} = \dfrac{2}{1}\]
Hence, \[BP:PD = 2:1\]. This means that the point \[P\] divides \[BD\] in the ratio \[2:1\] i.e., it divides \[BD\] into \[2 + 1 = 3\] parts.
Now, let us find \[BP\] and \[PD\]. Here, \[BP\] takes up 2 parts out of 3 parts. Similarly, \[PD\] takes up 1 part out of 3 parts.
It is given that \[BD:DC = 5:3\], i.e., \[BD\] is 5 parts. Thus,
\[BP = 5 \times \left( {\dfrac{2}{{2 + 1}}} \right) = \dfrac{{10}}{3}\]
\[PD = 5 \times \left( {\dfrac{1}{{2 + 1}}} \right) = \dfrac{5}{3}\]
Using the above quantities, we can find \[EG:GC\]. Let us consider $\vartriangle CEP$. In this triangle, we have \[EP\parallel AD\] i.e., \[EP\parallel GD\], since the point \[G\] lies on \[AD\].

We will apply Thales theorem to $\vartriangle CEP$. We have,
\[\dfrac{{GC}}{{EG}} = \dfrac{{DC}}{{PD}}\] ……….\[(1)\]
We know that \[DC\] is 3 parts and \[PD\] is \[\dfrac{5}{3}\] parts. Substituting these values in equation \[(1)\], we get
\[\dfrac{{CG}}{{}}\]\[\dfrac{{GC}}{{EG}} = \dfrac{3}{{\left( {\dfrac{5}{3}} \right)}}\]
Taking the denominator’s denominator to the numerator on the RHS, we get
\[\dfrac{{GC}}{{EG}} = \dfrac{9}{5}\]
Taking the reciprocal on both sides of the above equation, we get \[\dfrac{{EG}}{{GC}} = \dfrac{5}{9}\]. Hence, the required ratio is \[EG:GC = 5:9\].

Therefore, option C is the correct answer.

Note:
A triangle is a two-dimensional geometrical figure which has three sides. Thales theorem is also called Basic Proportionality theorem. The Basic Proportionality theorem can also be stated as “If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio”.