Quadrilaterals: Level 4 Challenges


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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