gmatt1476 wrote:
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 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
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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.
Makes sense. Thanks.