GMAT 750 Level Geometry DS Question 34

The given question is a challenging GMAT data sufficiency question testing concepts in Geometry. This GMAT 750 level DS question tests the following concepts in Geometry - Properties of Right Triangles, Applications of Pythagoras Theorem, and sides and angles properties of a triangle.

Question 34: In right triangle ABC, what is the ratio in which point D divides the hypotenuse AC? BD is perpendicular to AC.

Statement 1: BC = 2BD
Statement 2: AC = 2AB

Video Explanation

Explanatory Answer | GMAT Geometry Practice Question

Step 1: Evaluate Statement 1 ALONE

Statement 1: BC = 2BD

ABC is a right triangle. So are BDC and BDA.
In right triangle BDC, BC is hypotenuse.
If BC = 2BD, CD = \\sqrt{{BC}^{2} - {AC}^{2}}) = \\sqrt{{2BD}^{2} - {BD}^{2}}) = \\sqrt{{3BD}^{2}}) = √3 BD
So, the sides are in the ratio 1 : √3 : 2

In ΔBDC, angles opposite BD, CD, and BC will be 30°, 60° and 90°
So, we can deduce that ∠BAD = 60° and ∠DBA = 30°

ΔBDA is also a 30 – 60 – 90 triangle.
If BD = a, CD = √3 a and AD = \\frac{a}{\sqrt{3}})
CD : AD = √3 a : \\frac{a}{\sqrt{3}})
CD : AD = 3 : 1

We are able to find a UNIQUE answer to the question.
Hence, statement 1 alone is sufficient.
Eliminate answer option B, C, and E.

Step 2: Evaluate Statement 2 ALONE

Statement 2: AC = 2AB

If AC = 2 AB, sides of ΔABC will be in the ratio, AB : BC : AC = 1 : √3 : 2

So angles opposite AB, BC, and AC will be 30°, 60° and 90°
So, we can deduce that angle DBC = 60° and angle DBA = 30°

Because ΔBDA and ΔCDB are 30 – 60 – 90 triangles.
If BD = a, CD = √3 a and AD = \\frac{a}{\sqrt{3}})
CD : AD = √3 a : \\frac{a}{\sqrt{3}})
CD : AD = 3 : 1

We are able to find a UNIQUE answer to the question.
Hence, statement 2 alone is also sufficient.
Eliminate answer option A.

Statements are independently sufficient.

Choice D is the correct answer.