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
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked,but NEITHER statement ALONE is sufficient;
d)
EACH statement ALONE is sufficient to answer the question asked;
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
Correct answer is option 'D'. Can you explain this answer?

Question

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

a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked,but NEITHER statement ALONE is sufficient;
d)
EACH statement ALONE is sufficient to answer the question asked;
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct answer is option 'D'. Can you explain this answer?

Answer

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 =

= √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 = a√3
CD : AD = √3 a : a√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.

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 = a√3
CD : AD = √3 a : a√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.