In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex quadrilateral $ABCD$ is $180.$ Find the area of the shaded region.

ABJeIIy Jun 20, 2024

#1**+1 **

First, let's note that since the area of ABCD is 180, we have the equation

\(AD * AB =180\)

Now, note that the shaded region is a trapezoid with base AB and UT, with height AD.

We know the length of AB and AD already, and UT is 1/3 the length of DC, so we are all set.

We can write the equation

\((1/2)(AD)(AB + AB/3) = \\ (1/2)(AD)(4/3)(AB) = \\ (2/3) AD * AB = (2/3)(180) = 120\)

So our answer is 120.

Thanks! :)

NotThatSmart Jun 20, 2024

#1**+1 **

Best Answer

First, let's note that since the area of ABCD is 180, we have the equation

\(AD * AB =180\)

Now, note that the shaded region is a trapezoid with base AB and UT, with height AD.

We know the length of AB and AD already, and UT is 1/3 the length of DC, so we are all set.

We can write the equation

\((1/2)(AD)(AB + AB/3) = \\ (1/2)(AD)(4/3)(AB) = \\ (2/3) AD * AB = (2/3)(180) = 120\)

So our answer is 120.

Thanks! :)

NotThatSmart Jun 20, 2024