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Acute Angled Triangle...difference of two sides

I am totally clueless on this. I have tried Pythagoras theorem and tried taking a lot of ratios but don't seem to get anywhere:

The length of the sides of the acute angled triangle ABC are x-1, x and x+1. BD is perpendicular to AC. Then CD-DA equals

(a) x/8

(b) x/9

(c) 2

(d) 4

?

Re: Acute Angled Triangle...difference of two sides

OK guys I got it!

The answer is four.

First we equate

cos C = CD/(x+1) and cos C = [ (x+1)^2 + x^2 - (x-1)^2 ] / [2(x+1)(x)]

to find an expression for CD in terms of x then

cos A = AD/(x-1) and cos A = [ (x-1)^2 + x^2 - (x+1)^2 ] / [2(x-1)(x)]

to find an expression for DA in terms of x. Then subtracting CD-DA gives 4.

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Re: Acute Angled Triangle...difference of two sides

Quote:

Originally Posted by

**cosmonavt** OK guys I got it!

The answer is four.

First we equate

cos C = CD/(x+1) and cos C = [ (x+1)^2 + x^2 - (x-1)^2 ] / [2(x+1)(x)]

to find an expression for CD in terms of x then

cos A = AD/(x-1) and cos A = [ (x-1)^2 + x^2 - (x+1)^2 ] / [2(x-1)(x)]

to find an expression for DA in terms of x. Then subtracting CD-DA gives 4. **<--- correct! (Clapping)**

Here is a slightly different approach:

1. I've modified your sketch a little bit (see attachment)

2. Using Pythagorean theorem you'll get:

$\displaystyle (x-1)^2 - (x-k)^2 = (x+1)^2-k^2$

which simplifies to

$\displaystyle 2kx-2x=x^2+2x~\implies~k=\frac12 x +2$

3. Now determine using the result from #2

$\displaystyle k - (x-k)$

which yields 4.

Re: Acute Angled Triangle...difference of two sides

Yeah! That's way more simpler. Wonder why I didn't get that in the first place.