Area of trapezoid within a trapezoid

[themightypi]

Hi guys

The exercise is to find a general formula for the following problem. Imagine a trapezoid which height is divided into 4 equal parts, now parallel lines are drawn through these height markings (so 4 trapezoids are now within one big trapezoids, which all have the same heigt). Now the difficult part is, that I have to find a general formula on how to calculate the area of those 4 trapezoids.

any ideas? I know that the midsegment lines are proportional to each other.

thanks

 

[themightypi]

Im stuck to find a formula to search for the midsegment lines in each trapezoid. So the normal formula would be (a+c)/2. But now I cannot say how long the other midsegment lines of those 4 trapezoids are, because I do not know how much they increase/ decrease in relation to the one in the middle

 

[Steven G]

Please post a label diagram so we have something to talk about.

 

[Deleted member 4993]

Im stuck to find a formula to search for the midsegment lines in each trapezoid. So the normal formula would be (a+c)/2. But now I cannot say how long the other midsegment lines of those 4 trapezoids are, because I do not know how much they increase/ decrease in relation to the one in the middle

So you started with ONE trapezoid with height = 4h and parallel-sides of a & c

with the mid-line drawn you have two trapezoids of height each 2h (at this point I would have a sketch of the problem as I go along)

one trapezoid with parallel sides a & (a+c)/2 ....... median line of this trapezoid would be \(\displaystyle \frac{a + \frac{a+c}{2}}{2} = \frac{3a+c}{4}\)

second trapezoid with parallel sides c & (a+c)/2 ....... median line of this trapezoid would be \(\displaystyle \frac{c + \frac{a+c}{2}}{2} = \frac{3c+a}{4}\)

Now each of this would be cut into half again - each with height of 'h'

continue......

 

[Dr.Peterson]

Hi guys

The exercise is to find a general formula for the following problem. Imagine a trapezoid which height is divided into 4 equal parts, now parallel lines are drawn through these height markings (so 4 trapezoids are now within one big trapezoids, which all have the same heigt). Now the difficult part is, that I have to find a general formula on how to calculate the area of those 4 trapezoids.

any ideas? I know that the midsegment lines are proportional to each other.

Im stuck to find a formula to search for the midsegment lines in each trapezoid. So the normal formula would be (a+c)/2. But now I cannot say how long the other midsegment lines of those 4 trapezoids are, because I do not know how much they increase/ decrease in relation to the one in the middle

First, it's not quite accurate to say that the midsegment lines are proportional to each other; but we can say that they increase linearly, which may be of use to you.

But the formula you give is for the middle line of the three, right? Once you have that, aren't the others midsegments of the top and bottom halves?

 

[themightypi]

So you started with ONE trapezoid with height = 4h and parallel-sides of a & c

with the mid-line drawn you have two trapezoids of height each 2h (at this point I would have a sketch of the problem as I go along)

one trapezoid with parallel sides a & (a+c)/2 ....... median line of this trapezoid would be \(\displaystyle \frac{a + \frac{a+c}{2}}{2} = \frac{3a+c}{4}\)

second trapezoid with parallel sides c & (a+c)/2 ....... median line of this trapezoid would be \(\displaystyle \frac{c + \frac{a+c}{2}}{2} = \frac{3c+a}{4}\)

Now each of this would be cut into half again - each with height of 'h'

continue......

Thank you so much! Now I see that I can just continue by using this: \(\displaystyle \frac{a + \frac{a+c}{2}}{2} = \frac{3a+c}{4}\) as the bottom line and now use c (or a; dependant on which one is at the top) as the top line. From this I can just use the same formula again an then I receive: \(\displaystyle \frac{7a+c}{8}\) which I can then multiply with the height h.

 

[themightypi]

First, it's not quite accurate to say that the midsegment lines are proportional to each other; but we can say that they increase linearly, which may be of use to you.

But the formula you give is for the middle line of the three, right? Once you have that, aren't the others midsegments of the top and bottom halves?

Sorry, yes of course you are right, I meant the middle line!