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.