Quant Question Of The Day: 86
Geometry
A triangle is drawn with integer sides. Two sides are of length x and y such that (xy − 7)² = x² + y². What is the ratio of all possible obtuse triangles to that of all possible triangles?
A) 2 : 3
B) 3 : 5
C) 2 : 5
D) 5 : 7
E) 2 : 7
(xy -7)² = x² + y²
x²y² + 49 – 14xy = x² + y²
x²y² + 36 – 12xy + 13 = x² + y² + 2xy
( xy -6)² +13 = ( x + y)²
( x + y)² – (xy -6)² = 13
[ x + y – xy +6 ] [ x + y + xy -6 ] = 13 × 1
x, y => 3,4
( 4-3) < Third side < (4+3)
1 < z < 7
Total 5 triangles
3,4 , 2 => obtuse Angled
3,4 , 3
3, 4 , 5
3, 4, 4
3,4, 6 => obtuse Angled
So , 2/5
Option (C)
C) 2:5
The equation is satisfied when x and y are 3 and 4.
Out of all the possible triangles with two of its lengths 3 and 4, the ratio of obtuse angled triangles to that of total possible triangles is 2:5.