How Long Until You Notice the Trick?

[00:00:00]Welcome back. Today we've got a really elegant geometry puzzle. At first, it doesn't look like there's enough information, but with a couple of clever observations, everything falls into place. Let's take a look at the figure.

[00:00:17]We have a square A B C D. Point E lies on the left side A D. Point F lies on the top side D.

[00:00:29]joining E, F, and B forms triangle E FB inside the square. We're given two side lengths. EB is 10 units. FB is 8 units and the angle at F is exactly 90°.

[00:00:49]Our goal is to find the side length of the square. Let's call that unknown length X. We'll start with triangle E FB.

[00:01:01]Since it's a right triangle, we can apply the Pythagorean theorem.

[00:01:08]E F2 plus FB 2= E^2.

[00:01:17]Substituting the numbers gives E F 2 + 64 = 100. Subtract 64 from both sides and we get e f2 = 36.

[00:01:33]Taking the square root we find e f = 6.

[00:01:40]Now let's look at point f. Since D F and F C lie on the same straight line, the three angles around that straight line add up to 180°.

[00:01:58]The angle inside triangle E FB is 90°.

[00:02:05]So the other two angles together must also total 90°.

[00:02:11]Let's call them alpha and beta.

[00:02:16]That means alpha plus beta equals 90°.

[00:02:22]Next, look at the small triangle E D F.

[00:02:28]It has a right angle at D. One of its acute angles is alpha. So, the remaining acute angle must be beta.

[00:02:39]Now look at triangle F CB.

[00:02:43]It also has a right angle this time at C. One acute angle is beta. So the remaining angle must be alpha.

[00:02:54]That means these two corner triangles have exactly the same three angles. So they are similar triangles.

[00:03:03]The hypotenuse of the left triangle is E F which is 6. The hypotenuse of the right triangle is FB which is 8. So the similarity ratio is 6 to 8 or 3.

[00:03:22]Now let's use that ratio.

[00:03:25]In the right triangle, the side opposite angle beta is c, which is the side of the square. So its length is x.

[00:03:39]In the left triangle, the side opposite angle beta is d.

[00:03:45]Therefore, df equals 34s of x.

[00:03:52]The entire top side of the square DC also has length x.

[00:03:58]So fc equals x - 34 of x which is 1/4 of x.

[00:04:11]Now we're ready for the final calculation. Look at right triangle F CB.

[00:04:18]Its legs are FC which is 1/4 of X and CB which is X. Its hypotenuse is 8.

[00:04:28]Applying the Pythagorean theorem, the quantity 1/4 of X^2 + X^2 = 8^2.

[00:04:40]That becomes 116th of x^2 + x^2 = 64.

[00:04:48]Combine the two terms we get 17 / 16 * x^2 = 64.

[00:04:57]Multiply both sides by 16 and divide by 17. So, x^2 = 64 * 16 / 17. Finally, take the square root of both sides. The square<unk> of 64 is 8. The square<unk> of 16 is 4. 8 * 4 is 32. So, the exact value of x is 32 / the square<unk> of 17. And that's our final answer.