Problem number 20. Peter drew a quarter circle with the center at each corner of the flag, with dimensions 12 cm by 9 cm, and colored the region formed, as shown. What is the length indicated by the question mark? First, let's draw variables on the diagram to indicate the quarter circle's radii. For example, we have A in the bottom left, B in the top left, and so on, in clockwise order. Now, let's try and write equations relating these variables to the values that we were given. For example, if we look at the longer sides of the rectangles, we see that from the top side, B plus C is equal to 12, and from the bottom side, A plus the question mark plus D is equal to 12. That means that if we add these two equations together, we will get A plus B plus C plus D plus the question mark equals 24. Now, let's do a similar thing on the left and right sides of the rectangle. On the left side, we get A plus B equals 9. On the right side, we get C plus D equals 9. So if we add these together, we get A plus B plus C plus D equals 18. But now, let's look at these two equations closely. The first equation, A plus B plus C plus D plus question mark equals 24, is almost the same as the second equation, A plus B plus C plus D equals 18. The only difference is that the first equation has a question mark, and the value of the right side increased by 6. And that means that the question mark has to equal 6, since that's the only change between these two equations. In other words, you can subtract A plus B plus C plus D from the first equation, and 18 from the right hand side of the first equation, and you will get that the question mark equals 24 minus 18, or 6. And since the question asked us, what is the length indicated by the question mark, the answer is B, 6 centimeters.

geometry

quarter circles

rectangle

equations

length calculation