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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.

 

 Jun 20, 2024

Best Answer 

 #1
avatar+1252 
+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! :)

 Jun 20, 2024
 #1
avatar+1252 
+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

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