Question

Find the missing side for the Ceva’s theorem, \[\dfrac{{AF}}{{FB}} \cdot \dfrac{{BD}}{{DC}} \cdot \dfrac{{CE}}{X} = 1\]. Find X.

Answer 1

Hint: Ceva’s theorem states that if three lines are drawn in a triangle from each vertex to opposite sides, then the sides are divided equally or the lines will act as the bisector for the other side. Moreover, all the three lines intersect in a single point if the sides are divided into parts.
In this question, we need to determine the side from the given triangle which will replace the value of x. Moreover, it has already been mentioned in the question to use Ceva's theorem only.


Complete step by step solution:
According to Ceva’s theorem three lines are drawn in a triangle from each vertex to opposite i.e. from vertex A line AD is drawn on BC, from vertex B line BE is drawn on AC and from vertex C line CF is drawn on AB. So by the statement of Ceva’s theorem, these sides are divided into parts.

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As we can see that the point D divides the line BC; point F divides the line AB so, following Ceva's theorem, we can say that the point E will certainly divide the line AC. So,
\[\dfrac{{AF}}{{FB}} \cdot \dfrac{{BD}}{{DC}} \cdot \dfrac{{CE}}{X} = 1\]
Hence, X = EA
By statement of Ceva’s theorem, we can say that: \[\dfrac{{AF}}{{FB}} \cdot \dfrac{{BD}}{{DC}} \cdot \dfrac{{CE}}{{EA}} = 1\]


Note: For doing this question, students must have some knowledge of Ceva’s theorem so that they will be able to solve this. Moreover, students should be aware while using the sides in the numerator and the denominator.