A D , B E and C F are the medians of the triangle A B C . If A D ⊥ B E , then C F A B can be expressed as q p where p and q are co-prime positive integers. What is p + q ?
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What is. Apollonius' theorem
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Apollonius' theorem states that if A D is a median of △ A B C ,
The area of the green+ The area of the blue= The area of the red
In other words, 2 B D 2 + 2 A D 2 = A B 2 + A C 2 .
Hope this helps! More info here .
EDIT: Did you have a solution that doesn't use Apollonius' theorem? I'd like to see it.
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Consider the triangle defined by the coordinates ( 2 , 0 ) , ( 0 , 2 ) , and ( − 2 , − 2 ) . Easy solve? Perhaps. Cheating? Definitely ⌣ ¨
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@Daniel Liu – I agree.
@Daniel Liu – Did the same. Take the axes as the medians and take the triangle isosceles.
I used same method but I never new it was called Apollonius' theorem but thanks now that I have read the link you gave me.
I did it in exactly the same way but made that formula for median on my own and I didn't know that it was called Appolonius Theorem.
Let G be the point where AD meets BE. Then, Triangle AGB is a rectangular triangle. Then, as F is midpoint of AB, GF=AF=BF. By means, AB = 2GF.
On the other hand, CG=2GF (it's a properte of the Medians), then CF = 3GF.
Therefore, AB/CF = 2GF/3GF = 2/3 = p/q. Then, p = 2 and q = 3 ->> p + q = 5
same method mine :)
Decent one
If D is the midpoint of A B and G is the centroid of A B C , since B G A is a right triangle, A B G D = 2 1 . Since C D is a median, C D G D = 3 1 , hence the ratio is two thirds.
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We'll need to know the following things for this solution:
1 . The Pythagorean Theorem;
2 . The fact that the medians of a triangle trisect one another;
3 . Apollonius' theorem.
We're going to call the centroid [this is the point where the medians intersect one another] G for this problem. Also let A B = c , B C = a and C A = b for the sake of convenience.
Now notice that if A D = 3 x , then A G = 2 x and D G = x .
Similarly, if B E = 3 y , then B G = 2 y and E G = y .
Now apply the Pythagorean Theorem on triangles A G B , B D G and A G E to get the following equations respectively:
c 2 = 4 x 2 + 4 y 2 ⋯ ( 1 )
4 a 2 = 4 y 2 + x 2 ⋯ ( 2 )
4 b 2 = 4 x 2 + y 2 ⋯ ( 3 )
Add ( 2 ) and ( 3 ) together to get
4 a 2 + b 2 = 5 x 2 + 5 y 2
So, a 2 + b 2 = 5 ( 4 x 2 + 4 y 2 ) = 5 c 2 [See ( 1 ) ].
Now use Apollonius' theorem on △ A B C to get
a 2 + b 2 = 2 ( 4 c 2 + C F 2 ) .
That means 5 c 2 = 2 c 2 + 2 C F 2
And from that, C F 2 c = 9 4 = 3 2 .
So, p + q = 2 + 3 = 5 and that is our answer.