Question number 14. The circles in the picture are concentric and chord AB over here of length 16 is tangent to the smaller circle. What is the area of the region shaded here in blue? So let's label the radii here. We will need them to compute our area. So from the center of the circle, we have a radius of the larger circle here extending to the endpoints AB of that chord. So this will be labeled capital R. And the smaller radius here of the white circle that's not shaded, I'll call that lowercase r. And then the area of the shaded region the shaded region has area given by the difference of the areas of those circles. So pi capital R squared is the area of the larger circle minus pi lowercase r squared is the area of the smaller circle. And so we can simplify that by factoring out a pi and we have capital R squared minus lowercase r squared. So what we need is a relationship between these two. What is capital R squared minus lowercase r squared? So we go back to our picture. We can label some of the missing information here. We have that the chord AB has length 16 and it looks like we're looking at a triangle which is not necessarily a right triangle but we can create a right triangle over here. At the point of tangency of the chord to the smaller circle, we will draw a line from that point to the center along the lower, the smaller radius r. And that will guarantee us a right angle over here because anything tangent to the circle is perpendicular to the radius. So we have just drawn in another radius and what we're looking at now is a triangle, a right triangle which I will copy here. It has a hypotenuse that is the radius of the larger circle and then it has here these two legs which I will label. So the hypotenuse is capital R. One leg is half of 16 because we have just bisected that chord and the other leg is lowercase r. So now by the Pythagorean theorem what we have is capital R squared is equal to 8 squared minus r squared, excuse me, plus r squared and we can solve here for the quantity we're interested in. So moving things over to the left-hand side we have r squared minus lowercase r squared is 8 squared and so basically now we can we can substitute our quantity here in the equation we were working with initially and that gives us here finally pi times 8 squared and that's of course the same thing as 64 pi and that is then the expression we were looking for that gives us the area of the shaded region and so that is answer C here, 64 pi.

concentric circles

shaded region

tangency

Pythagorean theorem

area calculation