Area of Triangles in Trapezoid

• Jun 18th 2011, 06:42 PM
Area of Triangles in Trapezoid
Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

Here's a picAttachment 21694
• Jun 18th 2011, 07:10 PM
skeeter
Re: Area of Triangles in Trapezoid
Quote:

Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

Here's a picAttachment 21694

let $\displaystyle h_a$ = the perpendicular from O to AD

$\displaystyle h_b$ = the perpendicular from O to BC

$\displaystyle a = \frac{1}{2}h_a \cdot |AD|$

$\displaystyle b = \frac{1}{2}h_b \cdot |BC|$

area of the trapezoid ...

$\displaystyle A = \frac{1}{2}(h_a+h_b)(|BC| + |AD|)$

... finish it.

http://www.mathhelpforum.com/math-he...oid-trap8a.jpg
• Jun 18th 2011, 07:42 PM
Re: Area of Triangles in Trapezoid
Isn't that the regular formula for the area of a trapezoid though? im confused
• Jun 18th 2011, 11:25 PM
earboth
Re: Area of Triangles in Trapezoid
Quote:

Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O <-- is this correct? , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

If and only if the text of the question is correct (see the highlighted part in red) then the trapezium is a rectangle and

$\displaystyle a = b$

and consequently the area of the quadrilateral is

$\displaystyle A = 4\cdot a = 4 \cdot b$
• Jun 19th 2011, 04:12 AM
Zellator
Re: Area of Triangles in Trapezoid
Quote:

Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

You probably meant centroid here, right?

The way skeeter did it,
I can only consider that he used a as triangle ACD and b as triangle BAC, not the way the graph indicates.
Moreover, that BC is equal to the height of ABC
and AD is equal to the height of ACD. Am I right?

I don't understand how can you use the diagonals to calculate the height and make it a regular trapezoid formula.
$\displaystyle h_a+h_b=H$?

I am confused also :)
• Jun 19th 2011, 07:33 AM
skeeter
Re: Area of Triangles in Trapezoid
Quote:

Originally Posted by Zellator
I don't understand how can you use the diagonals to calculate the height and make it a regular trapezoid formula.
$\displaystyle h_a+h_b=H$?

$\displaystyle h_a$ is the shortest distance from O to base AD

$\displaystyle h_b$ is the shortest distance from O to base BC

let $\displaystyle H$ = shortest distance from AD to BC through point O

$\displaystyle H = h_a+h_b$

I was under the impression that the bases were given values along with the given areas a and b ... solve for h_a and h_b in terms of the given areas and the bases and you can derive an expression for the area of the trapezoid.

$\displaystyle A = \left(\frac{a}{|AD|} + \frac{b}{|BC|} \right) \cdot (|AD|+|BC|)$

also, as pointed out by earboth ...

Quote:

For any trapezoid ABCD. with diagonals BD and CA and midpoint O ,
... no way O can be the midpoint of the respective diagonals. That would make ABCD a rectangle or a rhombus.
• Jun 19th 2011, 09:12 AM