MartinTao wrote:
The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet?
A. 26
B. 32
C. 40
D. 64
E. 80
Two ways..
(1) I would recommend use of choices as a priority in such a question.
Let the hypotenuse be option A, that is 26, so one leg = 26-16=10....
So the other side = \(\sqrt{26^2-10^2}=24\)
Thus, the area = (24*10)/2=120... TRUE
so answer A
(2) Algebraic method
Let the legs be x and y, so hypotenuse = x+16 and area xy/2=120 or xy=240
by Pythagorean theorem \((x+16)^2=x^2+y^2.....x^2+32x+256=x62+y^2.......y^2=32x+256\) substitute x= 240/y from xy=240
Thus, \(y^2=32*\frac{240}{y}+256.......y^3-256y=32*240\)
We are not asked cubic functions, that is power of 3, but we can use \(y(y^2-256)=32*240.......(y-16)(y+16)y=32*240=8*24*40=(24-16)*24*(24+16)\)
so y = 24, x= 240/24=10 and hypotenuse = \(\sqrt{24^2+10^2}=26\)
A
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