Problem Solving

fskilnik@GMATH 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

Answer: [spoiler]____(C)__[/spoiler]

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

Cheers,
Brent