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.

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

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

@ INR

**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__.

**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.

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