The given question is GMAT 650+ level data sufficiency question testing concepts in triangles, isosceles triangles, and properties of right triangles.

Question 17: Is the triangle ABC right angled at B an isosceles triangle?

**Statement 1**: All 3 sides are integers.

**Statement 2**: The square of the hypotenuse is twice the product of the other two sides.

INR

**Statement 1**: All 3 sides are integers.

The ratio of the sides of an isosceles right triangle is 1 : 1 : √2

Either the sides with ratio 1 or the side with ratio √2 is not an integer.

From Statement 1 we know all sides are integers. If all three sides are integers, such a right triangle cannot be isosceles.

We are able to answer the question a conclusive NO using statement 1 ALONE.

Statement 1 ALONE is SUFFICIENT.

__Eliminate answer options B, C, and D__.

**Statement 2**: The square of the hypotenuse is twice the product of the other two sides

Let the sides be x, y, and z with z as the hypotenuse.

By Pythagoras theorem: x^{2} + y^{2} = z^{2}

From statement 2: z^{2} = 2xy

⇒ x^{2} + y^{2} = 2xy

⇒ x^{2} + y^{2} - 2xy = 0

⇒ (x - y)^{2} = 0

⇒ x – y = 0 ⇒ x = y

If x and y are same, the triangle is an isosceles right triangle.

We are able to answer the question a conclusive YES using statement 2.

Each statement is INDEPENDENTLY sufficient to answer the question.

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