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11 Feb 2019, 12:51
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
43% (02:30) correct
57%
(02:36)
wrong
based on 75
sessions
History
Date
Time
Result
Not Attempted Yet
GMATH practice exercise (Quant Class 7)
If O is the center of the circle with diameter AB and T is a tangency point, what is the value of x?
(A) 10
(B) 15
(C) 18
(D) 20
(E) 22.5
If O is the center of the circle with diameter AB and T is a tangency point, what is the value of x?
(A) 10
(B) 15
(C) 18
(D) 20
(E) 22.5
Updated on: 17 May 2021, 07:38
Originally posted by BrentGMATPrepNow on 11 Feb 2019, 15:23.
Last edited by BrentGMATPrepNow on 17 May 2021, 07:38, edited 1 time in total.
Last edited by BrentGMATPrepNow on 17 May 2021, 07:38, edited 1 time in total.
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fskilnik
Since angles in a triangle add to 180°, we know that the missing angle = (90 - 2x)°
Aside: Notice that (90 - 2x)° + 90° + 2x° =180°
Since angles on a line add to 180°, we know that the missing angle = (90 + 2x)°
Aside: Notice that (90 - 2x)° + (90° + 2x°) =180°
Next, draw a line from the center to the point of tangency. One of our circle properties tells us that this line is PERPENDICULAR to the tangent line.
Now focus on ∆ATO (below)
Since OT and OA are radii, we know that OT = OA, which makes ∆ATO an ISOSCELES triangle.
This means the other angle is ALSO (90 - 2x)°
Next, since OT is PERPENDICULAR to the tangent line, the remaining angle must be 2x°
Aside: Notice that (90 - 2x)° + 2x° =90°
Finally, we'll focus on the green triangle.
Since angles in a triangle add to 180°, we can write: x° + 2x° + (90 + 2x)° = 180°
Simplify: 5x° + 90° = 180°
Solve to get: x = 18
Answer: C
RELATED VIDEO
General Discussion
11 Feb 2019, 18:21
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fskilnik
Hi, Brent! Thank you for your nice contribution and beautiful drawings!
Alternate approach: (Click in the figure to "zoom" it! All angles are measured in degrees.)
\(? = x\)
\(\Delta BTC\,\,:\,\,\,3x + 2x = 90\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {{90} \over 5} = 18\)
The correct answer is (C).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
