SOLUTION: http://prntscr.com/aal28i I found x. I need help finding y. I'm pretty sure I'm supposed to use this theorem, and if not this one some theorem that has to do with proportions. Not

Question 1023122: http://prntscr.com/aal28i
I found x. I need help finding y. I'm pretty sure I'm supposed to use this theorem, and if not this one some theorem that has to do with proportions. Not the Pythagorean theorem. I'm supposed to be able to use proportions.
http://prntscr.com/aal2iu
http://prntscr.com/aal9g6
1 out of the 3 theorems I'm supposed to use.
I solved for x with corollary 1. x=2.25 which is 9/4, but I need to find y.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this reference explains what is happening.

http://www.regentsprep.org/regents/math/geometry/GP12/LMeanP.htm

the altitude to the hypotenuse of a right triangle creates 3 similar triangles.

in your particular case, triangle ABC is similar to triangle ADB and triangle ABC is also similar to triangle BDC and triangle ADB is similar to triangle BDC.

in other words, all 3 triangles are similar to each other.

the following diagram shows this relationship that is caused by dropping a perpendicular to the hypotenuse of a right triangle.

$$$

this creates the mean proportions that you see in the reference.

in particular, you get:

x/3 = 3/4 and you get x/y = y/(x+4) and you get 4/5 = 5/(x+4).

you can solve for x using x/3 = 3/4 to get x = 9/4.

you can replace x in the other equations to get:

x/y = y/(x+4) becomes (9/4)/y = y/(9/4 + 4) and you get 4/5 = 5/(9/4 + 4).

since 9/4 + 4 is equal to 9/4 + 16/4 which is equal to 25/4, then you get:

(9/4)/y = y/(25/4) and you get 4/5 = 5/(25/4).

in the first of these ratios, you get (9/4)/y = y/25/4) becomes (9/4)*(25/4) = y^2 after you cross multiply, which becomes y^2 = (9*25)/(4*4) which becomes y = (3*5)/(4) which becomes y = 15/4.

in the second of these ratios, you get 4/5 = 5/(25/4) which becomes 4*(25/4) = 5*5 which becomes 25 = 25 which is true, as it should be.

you wind up with x = 9/4 and y = 15/4.

you can confirm these values are correct by using the pythagorean formula of hypotenuse squared equals the one leg squared plus the other leg squared.

since x is equal to 9/4 and y is equal to 15/4, then (9/4)^2 + 3^2 should be equal to (15/4)^2 in triangle BDC.

(9/4)^2 = 81/16.
3^2 = 9 which is equal to 144/16.
(15/4)^2 = 225/16.

144/16 + 81/16 = 225/16.

looks like pythagorean formula works for those values of x and y so they should be good.

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