<PRE><B>Question 536661</B>
Essentially this problem specifies the area of the triangle, {{{A=(1/2)bh = (1/2)63*36=1134}}} and two sides.&lt;P&gt;
When given the area and two sides of a triangle use Heron&#39;s Formula which states.&lt;P&gt;
{{{A^2=1285956=((a + b + c)(a + b - c)(a - b + c)(b - a + c))/16}}} where a, b and c are the three sides of the triangle.&lt;P&gt;
Call a=60cm, b=63cm and c the unknown side.&lt;P&gt;
Let&#39;s try to simplify that right side before substituting what we know for a and b.&lt;P&gt;
{{{(a + b + c)(a + b - c)=a^2+ab+ac+ab+b^2+bc-ac-bc-c^2=a^2+2ab+b^2-c^2=(a+b)^2-c^2}}}&lt;P&gt;
{{{(a - b + c)(b - a + c)=ab-b^2+bc-a^2+ab-ac+ac-bc+c^2=-a^2+2ab-b^2+c^2=c^2-(a-b)^2}}}&lt;P&gt;
Multiply both sides by 16 to remove the denominator on the right.  That makes the equation:&lt;P&gt;
{{{20575296=((a+b)^2-c^2)(c^2-(a-b)^2)}}}&lt;P&gt;
Substitute the values we know:&lt;P&gt;
{{{20575296=((60+63)^2-c^2)(c^2-(60-63)^2)=(15129-c^2)(c^2-9)}}}&lt;P&gt;
Set {{{x=c^2}}} and rewrite the equation.&lt;P&gt;
{{{20575296=(15129-x)(x-9)=-x^2+15138x-136161=20575296}}}&lt;P&gt;
Subtract 20575296 from both sides.&lt;P&gt;
{{{-x^2+15138x-20711457=0}}}&lt;P&gt;
Multiply both sides by -1.&lt;P&gt;
{{{x^2-15138x+20711457=0}}}&lt;P&gt;
Maybe the Rain Man could factor that, but we&#39;ll have to use the quadratic equation to find x, then take the square root of x to find c.&lt;P&gt;
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}} =&lt;P&gt; 
{{{(15138 +- sqrt((-15138)^2-4*1*20711457))/(2*1)}}} =&lt;P&gt;
{{{(15138 +- sqrt(229159044-82845828))/(2)}}}=&lt;P&gt;
{{{(15138 +- sqrt(146313216))/(2)}}}=&lt;P&gt;
{{{(15138 +- 12096)/(2)}}}=&lt;P&gt;
That gives {{{x=(15138+12096)/2}}} and {{{x=(15138-12096)/2}}}&lt;P&gt;
{{{x=13617}}} and {{{x=1521}}}&lt;P&gt;
So {{{c=sqrt(13617)=116.69}}}and {{{c=sqrt(1521)=39}}}&lt;P&gt;
Both will work.  To prove it, plug into another version of Heron&#39;s Formula which is:&lt;P&gt;
{{{A = sqrt(s(s-a)(s-b)(s-c))}}}&lt;P&gt;
Where a, b and c are the sides, and s is the semiperimeter and {{{s = (a+b+c)/2}}}&lt;P&gt;
For c=116.69 this is:&lt;P&gt;
{{{s = (60+63+116.69)/2=119.85}}}&lt;P&gt;
{{{A = 1134= sqrt(119.85(119.85-60)(119.85-63)(119.85-116.69))}}}=&lt;P&gt;
{{{A = 1134= sqrt(119.85(59.85)(56.85)(3.16))}}}=&lt;P&gt;
{{{A = 1134= sqrt(1286291.58)=1134)}}}&lt;P&gt;
For c=39 this is:&lt;P&gt;
{{{s = (60+63+39)/2=81}}}&lt;P&gt;
{{{A = 1134= sqrt(81(81-60)(81-63)(81-39))}}}=&lt;P&gt;
{{{A = 1134= sqrt(119.85(21)(18)(42))}}}=&lt;P&gt;
{{{A = 1134= sqrt(1285956)=1134)}}}&lt;P&gt;
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