In the figure above, points A, B, C, and D are collinear and AB, BC

Question

https://gmatclub.com/forum/download/file.php?id=50478 In the figure above, points A, B, C, and D are collinear and AB, BC, and CD are semicircles with diameters d1 cm, d2 cm, and d3 cm, respectively. What is the sum of the arc lengths of semicircles AB, BC, and CD, in centimeters? (1) d1:d2:d3 is 3:2:1. (2) The length of line segment AD is 48 cm. DS45771.01 2019-09-22_0512.png

uchihaitachi · Answer

We need to find the lengths of the semicircles AB,BC and CD. 
The circumference of AB = pi*d1/2
The circumference of BC = pi*d2/2
The circumference of CD = pi*d3/2
Sum = pi*d1/2 + pi*d2/2 + pi*d3/2
we essentially need the sum of d1+d2+d3.

Option A. 
The ratio is given so that is variable. 
d1+d2+d3 = 6k where k is a variable. 
 Not Sufficient.

Option B 
d1+d2+d3 = 48. 
This is what we are looking for.  Sufficient.

IMO, Option B is the answer.

dabaobao · Answer

Length of semicircles = π(r1 + r2 + r3) = ? => (r1 + r2 + r3) = ?

1) No length given => Not sufficient
2) d1 + d2 + d3 = 48 => 2 (r1 + r2 + r3) = 48
Sufficient

ANSWER: B

altairahmad · Answer

What is 'length' of a semi-circle ? Is this even a term ? By 'length' one can mean the circumference e.g 'Length of the arc'.

A few of DS questions in geometry seem to be RC questions.

MathRevolution · Answer

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Visit https://www.mathrevolution.com/gmat/lesson for details. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary. The question asks the value of \frac{{πd_1}}{2}+\frac{{πd_2}}{2}+\frac{{πd_3}}{2} = \frac{{π(d_1+d_2+d_3)}}{2} . Since we have d_1 + d_2 + d_3 = 48 from condition 2), we have \frac{{π(d_1+d_2+d_3)}}{2} = 24π . Thus condition 2) is sufficient. Condition 1) If d_1 = 3, d_2 = 2, d_3 = 1 , then we have \frac{{π(d_1+d_2+d_3)}}{2} = 3π . If d_1 = 6, d_2 = 4, d_3 = 2 , then we have \frac{{π(d_1+d_2+d_3)}}{2} = 6π . Since condition 1) does not yield a unique solution, it is not sufficient. Therefore, B is the answer.

altairahmad · Answer

Makes sense. Thanks.

GDT · Answer

MentorTutoring

Pls clarify what exactly does length of semicircles means? Do we need to tell length of AD or circumference of semicircles

GDT · Answer

MentorTutoring

Pls clarify what exactly does length of semicircles means? Do we need to tell length of AD or circumference of semicircles

AndrewN · Answer

Hello,  GDT . In the question,  the lengths of semicircles AB, BC, and CD   must  refer to the circumference. Otherwise, you would expect to see the word  segment  in reference to segment AD; furthermore, peeking ahead at statement (2), when have you ever known a DS question to hand you the answer on a platter without any sort of work on your part? Even the easiest questions force you to manipulate the original expression or information to align (or not) with either of the two statements. As for this question, perhaps my GRE® cross-tutoring came in handy. I thought of  this question  from that test and walked away the answer here in well under a minute. It is a good exercise in not making assumptions.

Thank you for bringing my attention to the question.

- Andrew

GDT · Answer

MentorTutoring 
Thank you for the prompt reply!

My question was from PS perspective

And I wanted to confirm what it exactly meant as in either case statement 2 would have been sufficient

for circumference, pi d1/2 +pi d2/2 +pi d3/2 (length of curves)+ AD( length of straight line)= pi/2 (d1+d2+d3) + AD

suminha · Answer

We want to know exact length.
The ratio does not give any absolute value.

I highly recommend you to rephrase the question before jump into the choices.

Posted from my mobile device

Basshead · Answer

circumference of semi-circle =  \frac{πd}{2}

(1) We can't do anything with ratios. INSUFFICIENT.

(2) The length of AD = 48 cm.

\frac{48π}{2} = 24π

SUFFICIENT.

Answer is B.

Pritishd · Answer

1. The sum of the length of the semicircles will be  \frac{πd1}{2} + \frac{πd2}{2} + \frac{πd3}{2} + d1 + d2 + d3

2. Simplifying the above expression we get,  \frac{π}{2} (d1 + d2 + d3) + d1 + d2 + d3 = (d1 + d2 + d3) * (1 + \frac{π}{2})

3. We need the value of  d1 + d2 + d3  to answer the question

Statement - I ( Insufficient ) 
The ratio is not sufficient to derive the actual value of  d1 + d2 + d3

Statement - II ( Sufficient ) 
1.  AD  is  d1 + d2 + d3  =  48  cm
2. In other words we have have the value of  d1 + d2 + d3

Ans.  B

Bambi2021 · Answer

The sum of the circumferences of three circles with diameters a, b and c will be equal to the circumference of one circle with the diameter a+b+c.

B is sufficient.

GMATGuruNY · Answer

For the sake of clarity, the word in red should be included in the question stem:

Arc length of a semicircle =  \frac{πd}{2}

Statement 1:
Case 1: d₁ = 3, d₂ = 2, d₃ = 1
In this case, the sum of the arc lengths  = \frac{3π}{2} + \frac{2π}{2} + \frac{π}{2} = \frac{6π}{2} = 3π 
Case 2: d₁ = 30, d₂ = 20, d₃ = 10
In this case, the sum of the arc lengths  = \frac{30π}{2} + \frac{20π}{2} + \frac{10π}{2} = \frac{60π}{2} = ​30π 
Since the sum can be different values, INSUFFICIENT.

Statement 2:
Case 1: d₁ = 16, d₂ = 16, d₃ = 16
In this case, the sum of the arc lengths  = \frac{16π}{2} + \frac{16π}{2} + \frac{16π}{2} = \frac{48π}{2} = 24π 
Case 2: d₁ = 20, d₂ = 20, d₃ = 8
In this case, the sum of the arc lengths  = \frac{20π}{2} + \frac{20π}{2} + \frac{8π}{2} = \frac{48π}{2} = ​24π 
Since the sum in each case is the same, SUFFICIENT.

B

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In the figure above, points A, B, C, and D are collinear and AB, BC

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