Geometry

Quadrilaterals

Quadrilaterals: Level 4 Challenges

         

Suppose ABCDABCD is an isosceles trapezoid with bases ABAB and CDCD and sides ADAD and BCBC such that CD>AB.|CD| \gt |AB|. Also suppose that CD=AC|CD| = |AC| and that the altitude of the trapezoid is equal to AB.|AB|.

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

The numbers 3,4,3,4, and 66 denote the areas enclosed by their respective triangles.

What is the area of the yellow region?

On square ABCDABCD, points E,F,GE,F,G, and HH lie on sides AB,BC,CD,\overline{AB},\overline{BC},\overline{CD}, and DA,\overline{DA}, respectively, so that EGFH\overline{EG} \perp \overline{FH} and EG=FH=34EG=FH = 34. Segments EG\overline{EG} and FH\overline{FH} intersect at a point PP, and the areas of the quadrilaterals AEPH,BFPE,CGPF,AEPH, BFPE, CGPF, and DHPGDHPG are in the ratio 269:275:405:411.269:275:405:411. Find the area of square ABCDABCD.

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 ABCD is a parallelogram. Point PP on ABAB divides it in the ratio AP:PB=3:2 AP : PB = 3 : 2 , and point QQ on CDCD divides it in the ratio CQ:QD=7:3 CQ : QD = 7 : 3 . Let RR be the intersection of PQ PQ and ACAC. Then, AR:AC=a:b AR : AC = a : b , where aa and bb are positive coprime integers. What is a+ba + b ?

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