# Problem of the Day In the figure above, is tangent to circle at point . What is the area of the shaded region?

(1) The area of the circle is  .
(2) C. Both statements together are sufficient, but neither statement alone is sufficient.

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### Solution

[latexpage]

First, let’s make any deductions we can from the given information. Â 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. Â 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. Â To do this, the size of the circle must be defined and the area of the triangle must be defined. Â Neither of them is defined presently.

Statement #1: INSUFFICIENT. Â This statement defines the size of the circle. Â It allows you to find the radius by solving the equation $pi r^2=49$. Â  So, the circle is defined because the radius is defined. Â However, the area of the triangle can still vary. Â For example, we know nothing about $angle{B}$. Â 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. Â This statement allows us to fix the shape of the triangle. Â 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. Â With this information we know the relationship between the base and the height of the triangle. Â However, it tells us nothing about the size of the triangle or the size of the circle. Â 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. Â Taking both statements together fixes the size and shape, and thus the area, of the triangle as well as the size of the circle. Â This is enough to calculate the area of the shaded region by the method described above.