A rectangle with vertices 0,0, 100,0, 150, and 0,50 has a circle with center 75,30 and radius 10 cut out of it. What is the slope of the line through 75,30 that divides the remaining area of the rectangle into two equal parts? This is a little bit of a tricky question, but it has a clever trick. Let's draw the question that we're given on a coordinate grid, and note that any line through the center of the circle will necessarily bisect the circle. And let's try to think about why this must be true. If we draw any diameter of a circle, which we know will pass through the center of the circle, we know that it splits the circle into two exactly equal areas. Doing the same thing with the rectangle, if we draw any line through the center of the rectangle, it will always bisect the rectangle into two equal parts. It doesn't matter in what direction this line is drawn, but any straight line through the precise center of the rectangle will always split it in half. Using those pieces of information together, we can draw a line between the center of the circle and the center of the rectangle, and we know that this line will bisect the area of the rectangle minus the circle. This is a representation of what we'll have, and using this, we can very quickly find the slope of this line, and we find it to be 30-25 over 75-50, which we know is 1 5th, and our correct answer is A.

rectangle

circle

line slope

geometry problem

bisect area