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22 Nov 2021, 08:26
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AB, BC and CD are tangent to the yellow semicircle shown above. If O is the center of the circle (part of which the yellow semicircle is), the length of AB is 9 units, the length of CD is 16 units, and AO = OD, then what is the length of AD?
A. \(15\)
B. \(18\)
C. \(21\)
D. \(24\)
E. \(28\)
22 Nov 2021, 08:26
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Official Solution:
AB, BC and CD are tangent to the yellow semicircle shown above. If O is the center of the circle (part of which the yellow semicircle is), the length of AB is 9 units, the length of CD is 16 units, and AO = OD, then what is the length of AD?
A. \(15\)
B. \(18\)
C. \(21\)
D. \(24\)
E. \(28\)
The question has five exact answer choices (no "Cannot be determined" or "None of the above") so one of them must be correct. If one of the answers is correct, the answer must be correct no matter how we draw the diagram (of course we should not violate info that is given).
Re-draw the diagram so that points A and D coincide with the endpoints of the diameter. In this case, AB and DC will be perpendicular to the diameter. Check the new image below:
Draw perpendicular BF from point B to DC. CF will be \(DC - DF = 16-9=7\)
Next, when two tangent segments are drawn to one circle from the same external point, then they are equal, so \(BA = BG = 9\) and \(CD = CG = 16\). Therefore, \(BC = 9+16=25\).
In triangle, BCF, \(BC^2 = BF^2 + CF^2\)
\(25^2 = BF^2 + 7^2\)
\(BF=24\).
\(BF=AD=24\).
Answer: D
AB, BC and CD are tangent to the yellow semicircle shown above. If O is the center of the circle (part of which the yellow semicircle is), the length of AB is 9 units, the length of CD is 16 units, and AO = OD, then what is the length of AD?
A. \(15\)
B. \(18\)
C. \(21\)
D. \(24\)
E. \(28\)
The question has five exact answer choices (no "Cannot be determined" or "None of the above") so one of them must be correct. If one of the answers is correct, the answer must be correct no matter how we draw the diagram (of course we should not violate info that is given).
Re-draw the diagram so that points A and D coincide with the endpoints of the diameter. In this case, AB and DC will be perpendicular to the diameter. Check the new image below:
Draw perpendicular BF from point B to DC. CF will be \(DC - DF = 16-9=7\)
Next, when two tangent segments are drawn to one circle from the same external point, then they are equal, so \(BA = BG = 9\) and \(CD = CG = 16\). Therefore, \(BC = 9+16=25\).
In triangle, BCF, \(BC^2 = BF^2 + CF^2\)
\(25^2 = BF^2 + 7^2\)
\(BF=24\).
\(BF=AD=24\).
Answer: D
dbcoupie123
Joined: 30 Jan 2022
Last visit: 22 May 2025
Posts: 39
Given Kudos: 28
Location: Canada
Concentration: Finance
Schools: Yale '26 (A) Stern '26 (WL)
GMAT 1: 670 Q49 V32
GMAT 2: 690 Q46 V40
GMAT 3: 680 Q49 V34
GMAT 4: 710 Q48 V39
GPA: 3.7
WE:Operations (Finance: Investment Banking)
Updated on: 05 Feb 2023, 09:50
Originally posted by dbcoupie123 on 05 Feb 2023, 08:13.
Last edited by dbcoupie123 on 05 Feb 2023, 09:50, edited 1 time in total.
Last edited by dbcoupie123 on 05 Feb 2023, 09:50, edited 1 time in total.
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This solution and question does not make sense. By rearranging the tangent lines as provided in the explanation, you are effectively making line AD shorter to fit the diameter of the semicircle.
In the original figure, you can see that the line extends beyond the diameter of the semicircle. The answer/solution never accounted for that.
In the original figure, you can see that the line extends beyond the diameter of the semicircle. The answer/solution never accounted for that.
