Using the handout from today, we know Euclid's proof is based on similar triangles. With two similar triangles, the ratio between corresponding sides remains constant. If we list the sides of each triangle in the order short-leg-long leg-hypotenuse, here are the three triangles.
The big triangle: a-b-c
The triangle with b as hypotenuse: h-y-b
The triangle with a as hypotenuse: x-h-a
Using this information, given the lengths of one triangle we can find all the others.
Example: a-b-c are 3-4-5
The triangle with b as hypotenuse: h-y-b
h/3 = y/4 = 4/5
h = 12/5, y = 16/5
The triangle with a as hypotenuse: x-h-a
x/3 = h/4 = 3/5
x = 9/5, h = 12/5
Not surprisingly, we get the same value for h in both similar triangles it belongs to.
practice problems:
a) h-y-b are 3-4-5
b) x-h-a are 3-4-5
c) a-b-c are 8-15-17
Answers in the comments.
a) h-y-b are 3-4-5
ReplyDeletea-b-c triangle is a/3 = 5/4 = c/5
a = 15/4, c = 25/4
x-h-a triangle is x/3 = 3/4 = (15/4)/5
x = 9/4
b) x-h-a are 3-4-5
a-b-c triangle is 5/3 = b/4 = c/5
b = 20/3, c = 25/3
h-y-b triangle is 4/3 = y/4 = (20/3)/5 = 4/3
y = 16/3
c) a-b-c are 8-15-17
x-h-a triangle is x/8 = h/15 = 8/17
x = 64/17, h = 120/17
All we are missing is y. Another easier way to find it is x+y = c or y = c-x, so y = 17 - 64/17 = 225/17