Question 1079691: Let BAC be a right triangle. O is a point on side AC. D and E are points on side BC. AD is a line and OE is a line. DA intersects BO at point F. ADC is a right triangle. BOE is a right triangle.
Given AB = OA and OA equals CO, how do I calculate OF/OE?
This is so hard, my teacher said she doubts anyone can solve it. ;(
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let k be any positive number, so k > 0. The value k represents the distance from A to B, or the side length AB. Let's make
A = (0,0)
B = (0,k)
C = (2k,0)
O = (k,0) which is is on segment AC; specifically at the midpoint
D,E are points on segment BC; unknown locations for now
Notice how based on those definitions above, we would have
AB = k
AC = 2k
Furthermore, let's make these declarations:
line p = line through points A and D
line q = line through points B and O
line r = line through points O and E
line s = line through points B and C
We will use all four lines. They will pair up to help us locate points F and E. We do not need to know the location of D.
Line p and line q will pair up to find the coordinates of F
Line r and line s will pair up to find the coordinates of E
Yes there's a lot to remember here. I recommend drawing everything out to get a good idea of what things look like. See below
Note: Image generated using GeoGebra (free graphing software).
The red segments are the ones we truly care about, but will take a bit of work to find.
Take note of the letters next to each line. For instance, a "p" is next to the line through points A and D, which denotes that this is line p. Hopefully the labeling isn't too cluttered and you can find each label just fine.
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Let's get started finding the slope of line BC, which will help us find line p eventually.
slope of line BC:
m = (y2 - y1)/(x2 - x1)
m = (0-k)/(2k-0)
m = (-k)/(2k)
m = -1/2
Anything perpendicular to line BC will have a slope of 2/1, or just 2. We find this by flipping the fraction, and flipping the sign, of -1/2. This means that line p has a slope of 2.
line p has slope m = 2 and it goes through a known point (0,0), so we have (x,y) = (0,0) ---> x = 0 and y = 0
y = mx+b
y = 2x+b
0 = 2*0+b
b = 0
Therefore the equation of line p is y = 2x+0 or simply y = 2x
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Now let's find the equation for line q.
This line is going through two known points B = (0,k) and O = (k,0)
slope:
m = (y2 - y1)/(x2 - x1)
m = (0-k)/(k-0)
m = (-k)/(k)
m = -1
Use one of the points, say B = (0,k), and the slope and we get
y = mx+b
k = (-1)*0+b
k = 0+b
k = b
b = k
So y = mx+b turns into y = -x+k
If k was a small number, such as k = 1, then
y = -x+k
y = -x+1
which is the equation of line q.
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Now we have the equations
y = 2x
y = -x+1
Use any method you prefer to solve for (x,y) and you'll get (x,y) = (1/3, 2/3).
line p and line q intersect to form point F, so
F = (1/3, 2/3)
Keep this point in mind. The coordinates are very important.
Note: I skipped the steps on showing how I solved the system above to save time/space. Let me know if you need to see these steps.
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Now we need the equation for line r
We know the slope of line q is -1 which is the same as -1/1. Flip the fraction and sign and we get 1/1 = 1. The perpendicular slope is 1. The equation of line r will have slope of 1. Why? Because we want lines q and r to be perpendicular, so we can form a 90 degree angle BOE.
Point O is at (k,0) which is updated to (1,0) if we use k = 1 like above. It's important to stick with the same k value whatever it is you picked earlier.
Line r goes through (1,0) and the slope is 1, so
y = mx+b
y = 1x+b
0 = 1*1+b
0 = 1+b
0-1 = 1+b-1
b = -1
The equation for line r is: y = 1x-1 or just y = x-1
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We will use line r later. For now we need to find line s. This line goes through points B and C. Recall that I made k = 1, so
B = (0,k) = (0,1)
C = (2k,0) = (2*1,0) = (2,0)
We need the slope of the line through (0,1) and (2,0)
slope:
m = (y2 - y1)/(x2 - x1)
m = (0 - 1)/(2 - 0)
m = -1/2
Slope of line s = slope of segment BC = -1/2
B = (0,1) is the y intercept of line s, so that means the equation of line s is: y = (-1/2)x + 1
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Line r was found to be y = x-1
Line s was found to be y = (-1/2)x + 1
Use any method to solve and you should get (x,y) = (4/3, 1/3)
Note: I also skipped a few steps similar to the last system.
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At this point, you're probably wondering why I am doing all this work to find these coordinates. The reason I'm doing all this is so we can use the distance formula to find the distance between...
* Points F and O
* Points O and E
That will give the lengths of segments FO and OE respectively. Let's find the distance from F = (1/3, 2/3) to O = (1,0)
d = sqrt( (x1-x2)^2 + (y1-y2)^2 )
d = sqrt( (1/3-1)^2 + (2/3-0)^2 )
d = sqrt( 8/9 )
d = sqrt(8)/sqrt(9)
d = sqrt(8)/3
d = 0.94280904158207
The last decimal value is approximate
So FO = 0.94280904158207 approximately
Similarly, the distance from O = (1,0) to E = (4/3, 1/3) is...
d = sqrt( (x1-x2)^2 + (y1-y2)^2 )
d = sqrt( (1-4/3)^2 + (0-1/3)^2 )
d = sqrt( 2/9 )
d = sqrt(2)/sqrt(9)
d = sqrt(2)/3
d = 0.47140452079103
The last value shown above is approximate
So OE = 0.47140452079103 approx
Now there's one last step: dividing the lengths
FO/OE = 0.94280904158207/0.47140452079103 = 2.00000000000002
which is close enough. We have a bit of rounding error with the decimal approximations
If you ended up using the radical versions, with the square roots, then you'd perfectly end up with a ratio of 2.
FO/OE = (sqrt(8)/3)/(sqrt(2)/3) = (sqrt(8)/3) divided by (sqrt(2)/3) = 2
It's pretty interesting how all of this mess leads to a nice clean answer.
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Final Answer: 2
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