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December 14th, 2013, 11:04 PM   #1
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Geometry-Triangle

[attachment=0:1a0w2j4t]untitled.JPG[/attachment:1a0w2j4t]
Q: In a right triangle ABC,what is the maximum possible area of a square that can be inscribed when one of its vertices coincide with the vertex of the right angle of the triangle ?

(a)

(b)

(c)

(d)

Ansd). Now can this question be solved algebraically without using trigonometric ratios? Request a non-trigonometric solution.
Thanks in advance!
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December 14th, 2013, 11:28 PM   #2
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Re: Geometry-Triangle

Let the triangle above the square, which is similar to the larger triangle in which the square is inscribed have base and height .

The area of the square is then:



By similarity, we have:



We also have:



Hence:







And so:

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December 15th, 2013, 12:40 AM   #3
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Re: Geometry-Triangle

BC = a, AC = b, square sides = x; then AD = b - x and BF = a-x :
Code:
B

a-x

F      x        E

x               x

C      x        D        b-x          A
Since triangle BEF is similar to triangle EAD:
(a - x) / x = x / (b - x)
x^2 = (a - x)(b - x)
x^2 = ab - ax - bx + x^2
ax + bx = ab
x = ab / (a + b) ........area = x^2
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