Rectangle

[tulip frost]

Find the ratio of the area of the shaded region to the area of the rectangle.
O – the intersection of the diagonals of this rectangle.

[blazing pecan]

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[fringe verge]

what are ABCD?

[tulip frost]

i tried to do like this. I picked triangles with same areas

[fringe verge]

arbitrary points?

[lean mortar]

I think those are arbitrary.

[fringe verge]

but i assume AB=CD right

[lean mortar]

Yeah, that's the correct approach.

[lean mortar]

That is guaranteed due to O being the intersection of diagonals, I believe.

[fringe verge]

oh true

alr you got this cook 🔥

[tulip frost]

these are same. So from 2 of this kind of triangles both are shaded. so 2/2.

from 4 triangles that got same area 2 are shaded. which is 2/4

[lean mortar]

Right. Those two triangles are half of the rectangle.

[tulip frost]

and from 2 last (small) triangles 1 is shaded which is 1/2.

so 1/2 + 2/4 + 1 = 2

[lean mortar]

So, what's the shaded fraction?

[tulip frost]

but this is not correct

[tulip frost]

. this one isnt correct idk why

[lean mortar]

Suppose the whole rectangle has area 1.

What is the area of the two side triangles?

[lean mortar]

These ones, I mean.

[tulip frost]

0.25

cuz diogonals divide rectangle into 4 equal triangles.

[lean mortar]

Well, 1/4, yeah.

So, their total area is 1/2.

Now, let's get rid of them both. What's the total remaining area?

[tulip frost]

oh i got your point

[lean mortar]

I think your mistake was that here you considered it as the fraction of the whole rectangle instead of the remaining area.

[tulip frost]

ohhh my god exactly

[lean mortar]

So, it's not 1/2 + 2/4, but 1/2 + (2/4)(1/2), rather.

[tulip frost]

wait

remaining is 0.5 so whats not.

now we got 2 triangles. each is 0.25

[lean mortar]

The approach you can use here is just to use symmetry.

Here's the rectangle with the side triangles cut out.

Can you see that the white and gray parts are the same, just inverted around the center?

[tulip frost]

ooooooooohhhh

so it will be 0.25 + 0.5

[lean mortar]

Yeah, exactly!

[tulip frost]

+close

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