In triangle ABC, points D, E, F are on sides BC, CA, AB respectively such that AD, BE, CF are concurrent. If BD:DC = 2:3, CE:EA = 3:4, find AF:FB using Ceva's theorem.
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Step-by-Step Solution
Step 1: State Ceva's Theorem
Ceva's Theorem states that for a triangle ABC, if points D, E, F are on sides BC, CA, AB respectively, then the lines AD, BE, CF are concurrent if and only if the product of the ratios of the segments on each side is equal to 1. This theorem provides a powerful tool for solving problems involving concurrent lines in a triangle.
Step 2: Substitute Given Ratios
We are given the ratios BD:DC=2:3 and CE:EA=3:4. We substitute these values into Ceva's Theorem. The ratio BD:DC can be written as DCBD=32, and CE:EA as EACE=43.
Step 3: Simplify the Equation
Now, we simplify the product of the known ratios. Multiply the numerators and denominators: 2⋅3=6 and 3⋅4=12. This gives us 126, which simplifies to 21.
Step 4: Solve for AF:FB
To find the ratio AF:FB, we isolate it by dividing both sides of the equation by 21. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, 1÷21=1⋅2=2.