Easy Geometry Problem - Help Quick Please!

[boscord]

Check out this image/problem

Please just how me how to get the answer and provide it

Thanks in advance!

[thinktank2]

This problem is really easy to solve if you know there are similar triangles in the figure.



1) Triangle BOE and Traingle BDC are similar
2) Triangle COE and Triangle CAB are similar

Therefore ,

x/y = 12/(y+z) ----> equation 1

and x/z = 6/(y+z) ----> equation 2

so, x = 12*y/(y+z) = 6*z/(y+z)

which implies, 12*y = 6*z

or z=2*y

Substitute the value of z in equation 1

x = 12*y/(y + 2*y) = 12y/3y = 4

Therefore x=4

[prog_tom]

Extraordinary... Thinktank2, I'm impressed by your skills.
I have a question:

x/y = 12/(y+z) ----> equation 1 

and x/z = 6/(y+z) ----> equation 2

You said they are a system of linear equations, how do I solve it? Please be very specific. Thank you.

[thinktank2]

Originally posted by prog_tom
Extraordinary... Thinktank2, I'm impressed by your skills.
I have a question:

x/y = 12/(y+z) ----> equation 1 

and x/z = 6/(y+z) ----> equation 2

You said they are a system of linear equations, how do I solve it? Please be very specific. Thank you.

Solve what ? y and z ? (hint: read the problem's hint again)

The equations where derived using the property of the similar triangles
that they have two of their corresponding sides proportionate.
that is...

x/y = 12/(y+z) and x/z = 6/(y+z)

or x(y+z)-12y = 0 and x(y+z)-6z = 0

We have 2 equations with 3 variables.
To fix a definite value to all the 3 variables we need at least 3 equations.

From the 2 equations all we can say is x=4 and z = 2*y

From these it is pretty obvious that y or z can take any possible value.
That is, There are many solutions that satisfy the two equations.

[prog_tom]

Well ThinkTank2, with your breakthrough in this magnitude, I'm confident you will get anywhere you like.

You are a Princeton graduate?

[thinktank2]

[crptcblade]

Originally posted by thinktank2

Apparently Princeton only graduates people who know what triangles are. Those entrance exams must be killer!

[sql_lall]

What about the weirdest triangle of them all: _______
It has three angles, 0, 0, and 180 degrees....A.....B.....C
angle A = angle C = 0 degrees.

This is a triangle.