Quadrilateral

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Radit2125

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Could You help me solve this problem:
THE MIDDLES OF A QUADRANGLE ARE CONNECTED. WHAT IS THE RATIO BETWEEN THE AREAS OF THE TWO QUADRANGLES?
*the only thing I know is that the second quadrangle is a parallelogram
 

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Is it possible that you state clearly which you consider the 1st and 2nd parallelogram?
 
Radit2125 said:
Could You help me solve this problem:
THE MIDDLES OF A QUADRANGLE ARE CONNECTED. WHAT IS THE RATIO BETWEEN THE AREAS OF THE TWO QUADRANGLES?
*the only thing I know is that the second quadrangle is a parallelogram
Click to expand...
See attached image. I don't quite get your diagram.
In my image note the 4 shaded triangles. They appear to be congruent. What do you think? How many of them belong to the second quadrilateral? What's the ratio of areas?
 

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lev888 said:
See attached image. I don't quite get your diagram.
In my image note the 4 shaded triangles. They appear to be congruent. What do you think? How many of them belong to the second quadrilateral? What's the ratio of areas?
Click to expand...
Yes, I think also they are congruent and only 2 of them belong to the quadrilateral, but I don’t understand what is the ratio of areas.(is it possible to be 5:8)
 
Radit2125 said:
Yes, I think also they are congruent and only 2 of them belong to the quadrilateral, but I don’t understand what is the ratio of areas.(is it possible to be 5:8)
Click to expand...
First of all we need to figure out what's going on in your diagram. Which quadrilaterals are we talking about? I made my diagram based on your problem. But your diagram appears to contain another quadrilateral (in red). Please explain.

Regarding my diagram: the large quadrilateral is divided into 4 triangles. I picked one of them (shaded), but the same thing can be done with the other 3. We are asked about the ratio of the areas. So, in the shaded triangle there are 2 small triangles that belong to the 2nd quadrilateral. 4 belong to the first. What is the ratio of the areas?
 
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lev888 said:
First of all we need to figure out what's going on in your diagram. Which quadrilaterals are we talking about? I made my diagram based on your problem. But your diagram appears to contain another quadrilateral (in red). Please explain.

Regarding my diagram: the large quadrilateral is divided into 4 triangles. I picked one of them (shaded), but the same thing can be done with the other 3. We are asked about the ratio of the areas. So, in the shaded triangle there are 2 small triangles that belong to the 2nd quadrilateral. 4 belong to the first. What is the ratio of the areas?
Click to expand...

My diagram is wrong, but yours is not. Thank You for showing me an accurate diagram of the problem.
I suppose the ratio is 2:4 or 1:2. But is this the ratio of the areas of the 2 quadrilaterals? How should I describe the solution?
 
Radit2125 said:
My diagram is wrong, but yours is not. Thank You for showing me an accurate diagram of the problem.
I suppose the ratio is 2:4 or 1:2. But is this the ratio of the areas of the 2 quadrilaterals? How should I describe the solution?
Click to expand...
Yes, the ratio is 1:2 in the shaded triangle. But the same reasoning applies to the other 3 triangles. So, if the ratio is 1:2 in each triangle, what's the overall ratio of the 2 areas?
For a formal solution you should justify each step in my approach. Or use a different approach.
 
lev888 said:
Yes, the ratio is 1:2 in the shaded triangle. But the same reasoning applies to the other 3 triangles. So, if the ratio is 1:2 in each triangle, what's the overall ratio of the 2 areas?
For a formal solution you should justify each step in my approach. Or use a different approach.
Click to expand...
If the ratio is 1:2 in each triangle, does this probably mean that the overall ratio of the 2 areas is 2:1 (the sum of the ratios of all triangles).?
 
Radit2125 said:
If the ratio is 1:2 in each triangle, does this probably mean that the overall ratio of the 2 areas is 2:1 (the sum of the ratios of all triangles).?
Click to expand...
How did 1:2 become 2:1?
It's true, but "the sum of the ratios of all triangles" doesn't make sense. You can't just add ratios. There is a way prove it algebraically.
Quadrilateral area = area of 4 shaded triangles + 4 other triangles + 4 diff triangles + 4 more triangles
Parallelogram area = area of 2 shaded triangles + 2 ...
Ratio of the Parallelogram area to Quadrilateral area = ... = 1/2
Can you fill in what I skipped?
 
I would prefer to fill in what is missing in what You wrote:
Quadrilateral area = area of 4 shaded triangles + 4 other triangles + 4 diff triangles + 4 more triangles
Parallelogram area = area of 2 shaded triangles + 2 unshaded triangles
Ratio of the Parallelogram area to Quadrilateral area = 2:4= 1/2
 
Radit2125 said:
I would prefer to fill in what is missing in what You wrote:
Quadrilateral area = area of 4 shaded triangles + 4 other triangles + 4 diff triangles + 4 more triangles
Parallelogram area = area of 2 shaded triangles + 2 unshaded triangles
Ratio of the Parallelogram area to Quadrilateral area = 2:4= 1/2
Click to expand...
Quadrilateral area = area of 4 shaded triangles + 4 other triangles + 4 diff triangles + 4 more triangles = 4 ( ... )
Parallelogram area = area of 2 shaded triangles + 2 other triangles + 2 diff triangles + 2 more triangles = 2 ( ... )

What's inside the parentheses?
 
Yes, I am familiar with it. I made a mistake. In the space we should write a+ b+ c+ d
 
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