Tangent? Huh... what does that mean?
The tangent to a circle is a line that touches the circle at exactly one point. That point is known as the point of contact. GRE problem involving tangent almost always requires knowledge of this very important property:
The radius of the circle which joins the center of the circle to the point of contact (the tangent) is always perpendicular to the tangent.
In the figure above, line
is a tangent to the circle with center
at point of contact
. Segment
is a radius of the circle which is perpendicular to the tangent.
Example:
The above figure shows circle
with center
and a line
that is tangent to circle
at point
. If point
lies on line
such that
and radius of circle
is
, what is the area of the shaded region?
A) 
B) 
C) 
D) 
E) 
Solution:
From the figure we know that the shaded region represents
. To determine its area, we need to apply some special properties of the circle and the tangent. First lets organize the information provided:
Step 1: The problem states that line
is tangent to circle
at point
. Further we also know that point
lies on line
.
Therefore
. In addition we know that
is a radius of the circle
since
is the center and
is a point on the circumference of the circle. Since we know that radius of circle
is
, it follows that
. We can redraw the figure based on this information:
Step 2: From the figure we know that
is a right triangle with base
and height
. This is enough to calculate the area:
B is the correct answer.
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