A vertical line divides the triangle whose vertices are at (0,0), (1,1), and (9,1) into two parts; a triangle and a quadrilateral. If the areas of the triangle and the quadrilateral are equal, what is the equation of the vertical line?

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- Nov 8th 2010, 07:06 PMMATNTRNGTriangle divided...
A vertical line divides the triangle whose vertices are at (0,0), (1,1), and (9,1) into two parts; a triangle and a quadrilateral. If the areas of the triangle and the quadrilateral are equal, what is the equation of the vertical line?

- Nov 8th 2010, 08:13 PMpickslides
for vertical line make the areas of the shapes the same.

You can try to solve

All you have to do is determine which are linear combinations of the lines joining the triangle.

Can you find the equations of the 3 lines that form the triangle?

__Spoiler__: - Nov 8th 2010, 08:47 PMBAdhigraphical approach
from the diagram,

considering the area of the small triangle,find the area of it

If A is the area of the larger triangle,

substitute A, A1, (A+A') and A' with d and a

this is when vertical line meets AB

do the same thing when the vertical line meets CA - Nov 9th 2010, 10:59 AMWilmer
Ok, now that you've been shown how, try this similar one:

A vertical line divides the triangle whose vertices are at (0,0), (7,24), and (32,24) into two parts;

a triangle and a quadrilateral. If the areas of the triangle and the quadrilateral are equal,

what is the LENGTH of the vertical line?

You'll find that all side lengths (plus the areas) are integers;

smallest case, I believe, for an all-integer.

SOLUTION (since no response!):

A(0,0), B(7,24), C(32,24)

u=7,v=32,w=24 : so A(0,0), B(u,w), C(v,w)

Vertical line length = wSQRT[(v - u) / (2v)] = 15

Smallest all-integer case is smaller than I thought: A(0,0), B(4,15), C(36,15) ; vertical line = 10 - Nov 9th 2010, 12:11 PMpickslides
Might be easier to just find the area of the full triangle

Take half of that and solve for the smaller triangle section.

- Nov 9th 2010, 05:40 PMWilmer