lake triangle - need help

**sonia1** · December 22nd 2008, 12:02 PM

Determine how many acres there would be in that triangular lake, surrounded as shown by square plots of 370, 116 and 74 acres

**TheMasterMind** · December 22nd 2008, 12:15 PM

Originally Posted by sonia1

Determine how many acres there would be in that triangular lake, surrounded as shown by square plots of 370, 116 and 74 acres

$A=\sqrt{s(s-a)(s-b)(s-a)}$

$a=19.23538$

$b=10.77033$

$<br /> c=8.60233$

$s=\frac{a+b+c}{2} = \frac{\sqrt370 + \sqrt116 +\sqrt74}{2} = 19.30402$

$A=\sqrt{19.30402(0.06864)(8.53368)(10.70169)}=11.0 0037=11$

**sonia1** · December 22nd 2008, 12:26 PM

do u know how to work it out another way instead of using that formula..the long way

**TheMasterMind** · December 22nd 2008, 12:37 PM

Originally Posted by sonia1

do u know how to work it out another way instead of using that formula..the long way

$\frac{1}{4}$ $(\sqrt116+\sqrt74+\sqrt370)(\sqrt116+\sqrt74+\sqrt 370)^{1/2}$
$\frac{1}{4}$ $(\sqrt116+\sqrt370-\sqrt74)(\sqrt74+\sqrt370-\sqrt116)^{1/2}$

Can you take it from here? This way is very long. Are you sure you need to long way?

**Soroban** · December 22nd 2008, 01:31 PM

Hello, sonia1!

There is another approach . . . It requires Trig . . .

Determine how many acres there would be in a triangular lake,
surrounded by square plots of 370, 116 and 74 acres.

Code:

                              *
                          *    *
             ___      *         *   __
            √116  *              * √74 
              *                   *
          *                        *
    A *     *     *     *     *     * 
                    √370

Use the Law of Cosines to find angle $A.$

. . $\cos A \:=\:\frac{(\sqrt{116})^2 + (\sqrt{370})^2 - (\sqrt{74})^2} {2(\sqrt{116})(\sqrt{370})} \:=\:0.994345617$

Hence: . $A \:\approx\:6.1^o$

Then the area of the triangle is given by:

. . $A \;=\;\tfrac{1}{2}(\sqrt{116})(\sqrt{370})\sin6.1^o \:=\:11.007643903 \;\approx\;11\text{ acres}$

**sonia1** · December 23rd 2008, 06:25 AM

hi thanks for your help

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lake triangle - need help

how do u prove it

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