Question 218772
If a right triangle has a area 180cm^(2) and hypotenuse 41 cm, find the length of the two legs.


Step 1.  The Area {{{A=ab/2=180}}} where a and b are legs of the triangle.


{{{b=360/a}}}


Step 2.  The Pythagorean Theorem is the sum of the squares of the legs (a and b) is equal to the square of the hypotenuse c


{{{c^2=a^2+b^2}}}


{{{c^2=a^2+(360/a)^2}}}


Step 3. Multiply by a^2 to both sides of the equation to get rid of denominator and c=41.


{{{41^2*a^2=a^2*a^2+360^2}}}


{{{a^4-1681a^2+360^2=0}}}  


Step 4.  Let x=a^2  to simplify equation as a quadratic


{{{x^2-1681x+129600=0}}}


*[invoke quadratic "x", 1, -1681, 129600 ]


We have {{{x=1600}}} and {{{x=81}}},  then {{{a^2=1600}}} and {{{a^2=81}}}


Step 5. Taking the square root yields a=40 and a=9.


Then b is {{{360/40=9}}} or {{{360/9=40}}}.


Step 6.  ANSWER:  The legs of the right triangle are 9 cm and 40 cm.


I hope the above steps and explanation were helpful. 


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Also, good luck in your studies and contact me at john@e-liteworks.com for your future math needs.


Respectfully, 
Dr J


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