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Re: If O is the center of the circle with diameter AB and T is a tangency
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11 Feb 2019, 14:23
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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
Re: If O is the center of the circle with diameter AB and T is a tangency
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12 Feb 2019, 14:27
Expert Reply
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.
Re: If O is the center of the circle with diameter AB and T is a tangency
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22 Apr 2021, 15:38
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