Problem number 18. In the rectangle ABCD, the points E and F are marked on side DC as shown, so that angle EBA equals angle DFA equals 45 degrees, and AB plus EF equals 20 centimeters. What is the length of BC? To solve this problem, we'll use a lot of facts from geometry. So first, let's try to figure out what AB plus EF is equal to. Since we know that the problem tells us it's 20 centimeters, we can rewrite this length in a few ways. For example, we can write this as DC plus EF, because ABCD is a rectangle, and AB is equal to DC. Now, let's split up the length of DC into three parts, DE, EF, and FC. We can write DC plus EF equals DE plus EF plus FC plus EF. Now, let's rearrange the parentheses as shown. This is just the associative property of addition. We haven't done any geometry here. We've just moved the parentheses, so that now we have DE plus EF plus FC plus EF. And now, let's rewrite these two parts. DE plus EF, if we look at the diagram, is just DF, that length from D to F. Similarly, FC plus EF, if we add those together, we get the length from E to C. F is in the middle of E and C, so that's why we get EC. So that means that the value we have is equal to FD plus EC. And now, we'll justify this later, but I'm going to claim that FD plus EC is equal to AD plus BC, and we will prove this in a second. Even just looking at the diagram, you might be able to judge that FD is equal to AD, and EC is equal to BC, but we will prove this in a second. If you believe that, then we have the value currently equal to AD plus BC. But again, since this is a rectangle, AD and BC are the same length, so that means that this is equal to BC plus BC. So now, let's try to justify this step of why FD is equal to AD and EC is equal to BC. I've reproduced the diagram in a larger scale on the right, and I've marked the two angles that we're given. We know that the angle on a rectangle is 90 degrees. So using this, if we look at triangle ADF, we know that one angle is 90 degrees, and another angle is 45 degrees. And since the angle is added to 180 degrees, we know that the final angle on this triangle has to be 45 degrees. But notice that this triangle has two angles that are 45 degrees. That is, if we look at this triangle, this triangle is isosceles, because the two angles of 45 degrees are equal. That is, AD is equal to FD, and that's how we prove that FD is equal to AD in our step in the list of equations on the right. By the same logic, or by symmetry, you can prove that EC is equal to BC, because again, you have this 45-degree angle at the two bases of the equilateral triangle at points E and B, and therefore, EC is equal to BC. So this will show that all of our steps are valid, and now we know that BC plus BC is equal to AB plus EF. Since BC plus BC is equal to AB plus EF, which is 20 centimeters, BC is half of this quantity AB plus EF, or 20 centimeters. And that means that BC is just 10 centimeters. So that means that the answer is D, 10 centimeters.

rectangle

geometry

isosceles

angles

length