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If O is the center of the circle with diameter AB and T is a tangency

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fskilnik
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If O is the center of the circle with diameter AB and T is a tangency [#permalink] New post   11 Feb 2019, 12:51
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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
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C
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BrentGMATPrepNow
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If O is the center of the circle with diameter AB and T is a tangency [#permalink] New post   Updated on: 17 May 2021, 07:38
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fskilnik wrote:
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


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

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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.
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If O is the center of the circle with diameter AB and T is a tangency [#permalink] New post   11 Feb 2019, 18:21
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fskilnik wrote:
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

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.
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Re: If O is the center of the circle with diameter AB and T is a tangency [#permalink] New post   12 Feb 2019, 15:27
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fskilnik wrote:
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


A radius drawn to a tangent point yields a RIGHT ANGLE:

In the figure above:
Radius OT forms a right angle at tangent point T.
OA and OT are radii and thus are equal.
Since OA=OT, opposite angles OAT and ATO are equal.

We can PLUG IN THE ANSWERS, which represent the value of x.
From left to right, option D yields the following 3 triangles:

In this case, the resulting angle sum for the green triangle is TOO BIG:
20+80+90 = 190.
Eliminate D.

From left to right, option B yields the following 3 triangles:

In this case, the resulting angle sum for the green triangle is TOO SMALL:
15+60+90 = 165.
Eliminate B.

Since D yields a sum that is TOO BIG, while B yields a sum that is TOO SMALL, the correct answer must be BETWEEN B AND D.

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C
.
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Re: If O is the center of the circle with diameter AB and T is a tangency [#permalink]
12 Feb 2019, 15:27
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