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vipulgoyal
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I'm trying to put together the actual solution to find AD just for reference, but regardless the below question can be solved without doing that (and you should aim to solve similar questions without finding the actual answer, but rather just find out if it can be solved)
We have a triangle with essentially 6 segments of various lengths: AB,AC,BC (main triangle), AD (the line through it which is also the height),BD,CD (that make up BC).
Now realistically, we only really have four variables: AB,AC,AD,BD
This is because CD = 3 and BC = BD + CD = BD + 3, so we can substitute for those two.
In order to solve any equation with four variables we need four equations (and in general - to solve an equation with X variables, we'd need X dissimilar equations). We currently have three, because we have three right triangles:
1. AC^2 + AB^2 = BC^2 --> you can replace that with AC^2 + AB^2 = (BD+3)^2
2. AD^2 + 3^2 = AC^2
3. AD^2 + BD^2 = AB^2 (we know angle ADB is a right angle because angle ADC is a right angle)
Therefore, all we need is one more equation, or one value for either of the variables in our equations. Since either choice gives us a value for one of these variables, the answer is indeed D
