Blazing interpretation
In triangle A B C , ∠ B = 9 0 ∘ , B E = 3 , and B D = 4 .
If A E = E D = D C , then the value of A C can be expressed as a b , where a and b are positive integers and b is square-free.
Find a + b .
The answer is 8.
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3 solutions
did the same way!!
Exactly same way.
How did you get 23???
Solve this pair of equations to find A B and B C
( 3 2 A B ) 2 + ( 3 1 B C ) 2 = 3 2
( 3 1 A B ) 2 + ( 3 2 B C ) 2 = 4 2
to get A B = 2 3 , B C = 3 3 , which then leads to A C = 3 5
Sir you can just add those equations to get
9 5 A B 2 + 9 5 B C 2 = 2 5 .
9 5 A C 2 = 2 5
Then A C = 3 5
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we can just take center of AC to be the origin. Then if AC is d then a circle passing through ABC has radius as d/2. E=[d/6,0] and D[-d/6,0] . using this and equation of circle and finding distances EB and DB gives AC as required.
Can you explain how did you get this pair of equations?
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Coordinate Geometry. Take B as origin. A as (0,y); C as (x,0)
Which theory
let F be the midpoint of DE and AE=ED=CD = X then , BF = 3X/2 as in a right angle triangle median on hypotenuse is half of it. now use stewarts theorem in traingle BED considering BF to be the cevian and solve for x
From Apollonius' Theorem , we get
A B 2 = 2 + 2 E D 2 __(1)
B C 2 = 2 3 + 2 E D 2 __(2)
Add both equations we get
9 E D 2 = 2 5 + 4 E D 2
E D = 5
Which gives C A = 3 5 ~~~