Question 212048
The height of a triangle is 5 centimeters longer than three times its base. Find the base of the triangle if its area is 6 square centimeters


Step 1. Area of Triangle = {{{A = xh/2 =6}}}


where x=base,  h=height and  A=area of is one-half times base times height


Step 2.  {{{h=3x+5}}} (The height of a triangle is 5 centimeters longer than three times its base)


Step 3. Substitute height in Step 2 into equation in Step 1.


{{{A=xh/2=x(3+x5)/2}}}

{{{(3x^2+5x)/2=6}}}


Step 4.  Multiply 2 in both sides of last equation in Step 3 to get rid of denominator.


{{{2(3x^2+5x)/2=2*6}}}

{{{3x^2+5x=12}}}

Simplify by subtracting 12 from both sides of equation yields


{{{3x^2+5x-12=0}}}


Step 5.  The last equation of Step 4 is simply a quadratic equation.  Then use the quadratic formula:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}


where a=3, b=5, and c=-12


*[invoke quadratic "x", 3, 5, -12 ]


Step 6.  Pick the positive number in the quadratic equation.  In this case, we have the base x= 1.3333=4/3.  


height is then {{{h=3x+5=3(4/3)+5=9}}} centimeters

Area is then {{{(1/2)(4/3)(9)=6}}} square centimeters.  This is the same answer to our given answer.  So it check out.


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