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These shapes have four sides and 360° of interior angle goodness.

#### Challenge Quizzes

Suppose $$ABCD$$ is an isosceles trapezoid with bases $$AB$$ and $$CD$$ and sides $$AD$$ and $$BC$$ such that $$|CD| \gt |AB|.$$ Also suppose that $$|CD| = |AC|$$ and that the altitude of the trapezoid is equal to $$|AB|.$$

If $$\dfrac{|AB|}{|CD|} = \dfrac{a}{b},$$ where $$a$$ and $$b$$ are positive coprime integers, then find $$\large a^{b}.$$

The numbers $$3,4,$$ and $$6$$ denote the areas enclosed by their respective triangles.

What is the area of the yellow region?

On square $$ABCD$$, points $$E,F,G$$, and $$H$$ lie on sides $$\overline{AB},\overline{BC},\overline{CD},$$ and $$\overline{DA},$$ respectively, so that $$\overline{EG} \perp \overline{FH}$$ and $$EG=FH = 34$$. Segments $$\overline{EG}$$ and $$\overline{FH}$$ intersect at a point $$P$$, and the areas of the quadrilaterals $$AEPH, BFPE, CGPF,$$ and $$DHPG$$ are in the ratio $$269:275:405:411.$$ Find the area of square $$ABCD$$.

I want to make a litter box for my newly adopted pet kitten Admiral. The box is a cuboid with the top removed. I want the volume of the box to be 32 but the surface area to be minimized.

What is the minimum internal surface area I can achieve?

$$ABCD$$ is a parallelogram. Point $$P$$ on $$AB$$ divides it in the ratio $$AP : PB = 3 : 2$$, and point $$Q$$ on $$CD$$ divides it in the ratio $$CQ : QD = 7 : 3$$. Let $$R$$ be the intersection of $$PQ$$ and $$AC$$. Then, $$AR : AC = a : b$$, where $$a$$ and $$b$$ are positive coprime integers. What is $$a + b$$?

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