Section: Math Section 2
37)
In the figure above, BC = 4, CD = 5, and AD = 12. If point E lies somewhere between points B and C on line segment BC, what is one possible length of AE?
Explanation
The first thing to do is to put the numbers 4, 5, and 12 in the appropriate places in the figure. Now you should see that you have the lengths of 2 sides of triangle ACD. Since ACD is a right triangle, you can use the Pythagorean theorem to figure out the length of the hypotenuse, but if you’ve memorized the common Pythagorean triplets you don’t have to do that—you’ll immediately recognize that this is a 5-12-13 right triangle, and so the length of AC is 13. The length of BD is 4 + 5 = 9, so the triangle ABD has legs of lengths 9 and 12. Again, you can use the Pythagorean theorem to find the length of the hypotenuse, but you should notice that ABD is a multiple of the 3-4-5 right triangle, and AB has length 15. If you draw in point E in the figure between B and C, you’ll see that AE will be longer than AC but shorter than AB, or greater than 13 but less than 15. So, any number between 13 and 15, such as 14, is a possible answer.
