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

by fskilnik@GMATH » Mon Feb 11, 2019 11:49 am

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GMATH practice exercise (Quant Class 7)

Image

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

Answer: [spoiler]____(C)__[/spoiler]
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by Brent@GMATPrepNow » Mon Feb 11, 2019 2:11 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 7)

Image

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

Answer: [spoiler]____(C)__[/spoiler]
Since angles in a triangle add to 180°, we know that the missing angle = (90 - 2x)°
Image
Aside: Notice that (90 - 2x)° + 90° + 2x° =180°


Since angles on a line add to 180°, we know that the missing angle = (90 + 2x)°
Image
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.
Image


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)°
Image


Next, since OT is PERPENDICULAR to the tangent line, the remaining angle must be 2x°
Image
Aside: Notice that (90 - 2x)° + 2x° =90°


Finally, we'll focus on the green triangle.
Image
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

Cheers,
Brent
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If O is the center of the circle with diameter AB and T is a

by fskilnik@GMATH » Mon Feb 11, 2019 4:57 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 7)

Image

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
Post appeared twice. Please see my solution below!
Last edited by fskilnik@GMATH on Tue Feb 12, 2019 4:50 am, edited 1 time in total.
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If O is the center of the circle with diameter AB and T is a

by fskilnik@GMATH » Mon Feb 11, 2019 5:12 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 7)

Image

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!

Image

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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by GMATGuruNY » Tue Feb 12, 2019 2:12 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 7)

Image

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:
Image
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:
ImageImageImage
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:
ImageImageImage
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.

The correct answer is C.
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