Question 173140
Let b=base and h=height


Since "The height of a triangle is 8 centimeters greter than the base", this means that {{{h=b+8}}}


Also, remember that the formula for the area of a triangle is 


{{{A=(b*h)/2}}}


Since the "area of the triangle is 90 square centimeters", this tells us that {{{A=90}}}



{{{90=(b(b+8))/2}}} Plug in {{{A=90}}} (the given area) and {{{h=b+8}}}



{{{2*90=b(b+8)}}} Multiply both sides by 2



{{{180=b(b+8)}}} Multiply



{{{180=b^2+8b}}} Distribute



{{{0=b^2+8b-180}}} Subtract 180 from both sides. 




Notice we have a quadratic equation in the form of {{{ab^2+bb+c}}} where {{{a=1}}}, {{{b=8}}}, and {{{c=-180}}}



Let's use the quadratic formula to solve for b



{{{b = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{b = (-(8) +- sqrt( (8)^2-4(1)(-180) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=8}}}, and {{{c=-180}}}



{{{b = (-8 +- sqrt( 64-4(1)(-180) ))/(2(1))}}} Square {{{8}}} to get {{{64}}}. 



{{{b = (-8 +- sqrt( 64--720 ))/(2(1))}}} Multiply {{{4(1)(-180)}}} to get {{{-720}}}



{{{b = (-8 +- sqrt( 64+720 ))/(2(1))}}} Rewrite {{{sqrt(64--720)}}} as {{{sqrt(64+720)}}}



{{{b = (-8 +- sqrt( 784 ))/(2(1))}}} Add {{{64}}} to {{{720}}} to get {{{784}}}



{{{b = (-8 +- sqrt( 784 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{b = (-8 +- 28)/(2)}}} Take the square root of {{{784}}} to get {{{28}}}. 



{{{b = (-8 + 28)/(2)}}} or {{{b = (-8 - 28)/(2)}}} Break up the expression. 



{{{b = (20)/(2)}}} or {{{b =  (-36)/(2)}}} Combine like terms. 



{{{b = 10}}} or {{{b = -18}}} Simplify. 



So the possible lengths of the base are {{{b = 10}}} or {{{b = -18}}} 

  
  
However, a negative base does NOT make sense. So this means that the only possible length for the base is {{{b=10}}}



So the base is 10 centimeters.



{{{h=b+8}}} Go back to the height equation (remember the height is 8 cm longer than the base)



{{{h=10+8}}} Plug in {{{b=10}}}



{{{h=18}}} Add



So the height is 18 cm



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Answer:


So the base is 10 cm and the height is 18 cm.





Check:



Area: {{{A=(10*18)/2=180/2=90}}}