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The figure below shows the dimensions of the right triangle ABC with AB = 13, BC = 12, CA = 5 and I is a point inside tr

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by Max@Math Revolution » Mon Feb 03, 2020 12:27 am
[GMAT math practice question]

The figure below shows the dimensions of the right triangle ABC with AB = 13, BC = 12, CA = 5 and I is a point inside triangle ABC. Angle C is 90. What is the minimum distance from the point I to sides AB, BC and CA?
2.3ds.png
1) Point I is the incenter of △ABC.
2) Line AI bisects angle A, and line BI bisects angle B.
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Source: — Data Sufficiency |
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Re: The figure below shows the dimensions of the right triangle ABC with AB = 13, BC = 12, CA = 5 and I is a point insid

by Max@Math Revolution » Wed Feb 05, 2020 5:16 pm
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Condition 1)
2.3DS.A1.png
Since I is the incenter of the triangle, the distances to all sides from point I are equal. Assume the distances are x.
2.3DS(A2).png
The area of triangles IAB, IBC and ICA are (1/2)*13*x + (1/2)*12*x + (1/2)*5*x = (13/2)x + 6x + (5/2)x = (18/2)x + 6x = 9x + 6x = 15x.
The area of triangle ABC = (1/2)*5*12 = 30.
Since the sum of the areas of triangles IAB, IBC and ICA is equal to the area of triangle ABC, we have 15x = 30 or x = 2.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since we can find an incenter of a triangle by the intersection of lines bisecting interior angles, I is the incenter of the triangle from condition 2).

Thus, condition 2) is sufficient with the previous reasoning in condition 1).

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
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