$\"\"$

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The vertices of the trapezoid are $\"\"$ and $\"\"$ .

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Need to prove that the centroid of the trapezoid is the point of intersection of the line connecting the midpoints of the parallel sides and the line connecting the extended parallel sides.

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Draw the related diagram of trapezoid:

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$\"\"$

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Find the line equations of $\"\"$ and $\"\"$ .

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$\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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Equation of line $\"\"$ .

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Point slope form of a line equation is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$ .

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Equation of line $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Find the intersection point of $\"\"$ and $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The point of intersection is $\"\"$ .

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$\"\"$

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The area of the trapezoid is $\"\"$ .

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$\"\"$ .

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First find the curve equations:

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Equation of line $\"\"$ .

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The vertices of the trapezoid are $\"\"$ and $\"\"$ .

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Equation of line $\"\"$ .

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Point slope form of a line equation is $\"\"$ .

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$\"\"$

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$\"\"$ .

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Equation of line $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ and $\"\"$ .

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Find the $\"\"$ -coordinate of the centroid.

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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Find the $\"\"$ -coordinate of the centroid.

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Substitute $\"\"$ and $\"\"$ in $\"\"$ . $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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Therefore, centroid is $\"\"$ .

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$\"\"$

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$\"\"$ .