In pallelogram, ABCD, E is the midpoint of DC, AE intersecs BD at point O, the area of triangle DOE is 12cm^2, what is the area of triangle AOB?

This should help out:

http://www.jiskha.com/display.cgi?id=1317372273

To find the area of triangle AOB, we can use the fact that the area of a triangle is equal to half the product of its base and height.

In the given parallelogram ABCD, E is the midpoint of DC. This means that DE = EC.

We are told that AE intersects BD at point O. Let's denote the intersection point of AE and BD as point O.

Now, let's examine triangle DOE. We are given that the area of triangle DOE is 12 cm^2.

We know that the base of triangle DOE is DE, and the height is AO (the perpendicular distance from O to DE).

Using the area formula, we have: Area of triangle DOE = (1/2) * base * height
Substituting the given values, we get: 12 = (1/2) * DE * AO

Since DE = EC, we can rewrite the equation as: 12 = (1/2) * EC * AO

We know that DC is parallel to AB in a parallelogram, so triangle AOB is similar to triangle DOE by the AA similarity criterion.

By similarity, we have: AO/DE = BO/EO

Substituting AO = x and DE = EC, we get: x/EC = BO/EO

We can rewrite this equation as: x = (BO/EO) * EC

Now, let's substitute this value of x into the equation we derived earlier: 12 = (1/2) * EC * ((BO/EO) * EC)

Simplifying the equation, we get: 12 = (1/2) * (BO/EO) * EC^2

To find the area of triangle AOB, we need to find BO and EO.

To find BO, we can use the property that in a parallelogram, opposite sides are equal. As ABCD is a parallelogram, BC = AD. Also, we know that DC = 2EC (since E is the midpoint of DC). Therefore, BO = AD - DC = BC - DC = (BC - DC) = EC.

Similarly, EO = DC - DE = DC - EC = 2EC - EC = EC.

Substituting these values into the equation, we get: 12 = (1/2) * (EC/EC) * EC^2

Simplifying further, we have: 12 = (1/2) * EC * EC

Rearranging the equation, we get: EC^2 = 24

Taking the square root of both sides, we find: EC = √24 = 2√6

Now that we have EC, we can find BO and EO:
BO = EC = 2√6 cm
EO = EC = 2√6 cm

Thus, triangle AOB is an isosceles triangle with BO = EO = 2√6 cm.

Finally, let's calculate the area of triangle AOB using the formula: Area = (1/2) * base * height

The base of triangle AOB is BO, and the height is EO. Substituting the values, we have:
Area of triangle AOB = (1/2) * BO * EO = (1/2) * 2√6 * 2√6 = 2 * 6 = 12 cm^2

Therefore, the area of triangle AOB is 12 cm^2.