Problem of the Day


In the figure above, \overline{AC} is tangent to circle B at point A. What is the area of the shaded region?

(1) The area of the circle is 49 in^2.
(2) AB:BC=1:2

Reveal Answer

Answer

C. Both statements together are sufficient, but neither statement alone is sufficient.

See the Solution

Solution

[latexpage]

First, let’s make any deductions we can from the given information. ¬†The fact that $overline{AC}$ is tangent to circle $B$ at point $A$ implies that $angle{BAC}=90^{circ}$. Thus, $triangle{ABC}$ is a right triangle.

Now, think about what you would normally do to find the shaded area in a problem like this.  The typical way to find it is to first find the area of the triangle and then subtract the area of the sector of the circle.  To do this, the size of the circle must be defined and the area of the triangle must be defined.  Neither of them is defined presently.

Statement #1: INSUFFICIENT.  This statement defines the size of the circle.  It allows you to find the radius by solving the equation $pi r^2=49$.   So, the circle is defined because the radius is defined.  However, the area of the triangle can still vary.  For example, we know nothing about $angle{B}$.  If you vary the measure of $angle{B}$ from small to large, it is clear that the area triangle and hence the area of the shaded region changes:

 

Statement #2: INSUFFICIENT.  This statement allows us to fix the shape of the triangle.  The ratio, together with the fact that we already know $angle{BAC}=90^{circ}$, implies that $triangle{ABC}$ is a $30^{circ}-60^{circ}-90^{circ}$ right triangle.  With this information we know the relationship between the base and the height of the triangle.  However, it tells us nothing about the size of the triangle or the size of the circle.  We can make the area of the shaded region arbitrarily large by increasing the radius of the circle, or arbitrarily small by making it closer and closer to zero.

Statements 1 & 2: SUFFICIENT.  Taking both statements together fixes the size and shape, and thus the area, of the triangle as well as the size of the circle.  This is enough to calculate the area of the shaded region by the method described above.


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