Statement 1: The area of triangular region ABD is equal to the area of triangular region DBC.

This means that the point D is mid-point of AC. But, we can't still say whether the triangle ABC is isosceles or not. Insufficient.

Statement 2: BD┴AC and AD = DC. Signifies that the median drawn from the vertex B is a perpendicular bisector. This can only happen if the triangle is isosceles triangle with AB = BC. Sufficient

The answer should be B

The correct answer is B.
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It means that BD is perpendicular to AC.
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Collection of Questions:
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DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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In the figure above, D is a point on the side AC of ΔABC. Is ΔABC isosceles?

(1) The area of the triangular region ABD is equal to the area of triangular region DBC.

(2) BD is perpendicular to AC and AD = DC.

Answer: B

We are after: Is ΔABC isosceles?

(1) Not sufficient. Using the formula for areas of triangle, we get


Area of ΔABD = Area of ΔABD
1/2 * AD * h = 1/2 * DC * h

h is the height for both triangles, if you drop a perpendicular from point B on AC. This will fall on the extended AC (if you can imagine it)

The above equation yields to:
AD = AC

But we want to know if ΔABC is isosceles. This information doesn't help so it is INSUFFICIENT.

(2) Sufficient

Since BD ⊥AC, and D is on AC, imagine D dividing AC into two sides i.e. AD =AC.

Now imagine two right angled Δs inside ABC being split in the middle by BD. These two right angled Δs have the same base and height so the hypotenuse will be same i.e. AB = AC
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