# GMAT 650 Level Quant Question 17 | Geometry

#### GMAT Sample Questions | Data Sufficiency Practice

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.

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### Explanatory Answer | GMAT Geometry DS

#### Step 1: Evaluate Statement 1 ALONE

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.

#### Step 2: Evaluate Statement 2 ALONE

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: x2 + y2 = z2
From statement 2: z2 = 2xy
⇒ x2 + y2 = 2xy
⇒ x2 + y2 - 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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