Question 1103131
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The perimeter of a right triangle is 60 inches and the length of the altitude to the hypotenuse is 12 inches. 
How many square inches are in the area of the triangle?
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<pre>
Let x and y be the legs lengths.


Then the hypotenuse is  {{{sqrt(x^2+y^2)}}}.


The "perimeter" equation is

{{{x + y + sqrt(x^2+y^2)}}} = 60,         (1)

which implies

{{{sqrt(x^2+y^2)}}} = 60 - (x+y),

{{{x^2 + y^2}}} = {{{60^2 - 2*60*(x+y) + (x+y)^2}}},

{{{x^2 +y^2}}} = {{{3600 - 120*(x+y) + x^2 + 2xy + y^2}}},

2xy = 120*(x+y) - 3600,

xy = 60*(x+y) - 1800.      (2)


From the other side, you have this equation expressing the area of the right-angled triangle 

{{{(1/2)*xy}}} = {{{(1/2)*c*12}}},

which gives

xy = {{{12*sqrt(x^2+y^2)}}}.           (3)


In this equation, replace  {{{sqrt(x^2+y^2)}}}  in the right side by  [60 - (x+y)], based on  (2).


You will get instead of (3)

xy = 12*(60 - (x+y)).       (4)


Now compare equations (2) and (4).  Their left sides are identical; hence, right sides are equal:

60*(x+y) - 1800 = 12*(60 - (x+y)).


It implies  60*(x+y) + 12*(x+y) = 12*60 + 1800,  or 72*(x+y) = 2520,  and then  x+y = {{{2520/72}}} = 35.


Thus we found that the sum of the leg lengths is 35 inches.  


Then the hypotenuse is  60 - 35 = 25 inches  (from the condition on the perimeter),


and the area of the triangle, which is under the question, is  {{{(1/2)*25*12}}} = 25*6 = 150 square inches.
</pre>

SOLVED.


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By knowing that x+y = 35 and x^2 + y^2 = 25^2,  it is easy to find the values of x and y individually.


At this level, it is just standard exercise in solving quadratic equations, and I leave it to you.