Four Trapezia Formed by the Difference of Two Concentric Squares

ABCD is a square of side 10 cm. PQRS is a square inside ABCD. PQBA, QRCB, RSDC, and SPAD are identical trapezia, each of area 16 cm2. What is the height of each trapezium if PQ is parallel to AB and SR is parallel to DC?

A.   3 cm C.   2 cm
B.   1.8 cm D.   1.2 cm



Common Quadrilaterals




Area, $A = a^2$

Perimeter, $P = 4a$

Diagonal, $d = a\sqrt{2}$


01 Area enclosed by line rays inside a square

The figure shown below is a square of side 4 inches. Line rays are drawn from each corner of the square to the midpoints of the opposite sides. Find the area of the shaded region.

Four-pointed star inside the square


10 Area common to three squares inside the regular hexagon

Three squares are drawn so that each will contain a side of regular hexagon as shown in the given figure. If the hexagon has a perimeter of 60 in., compute the area of the region common to the three squares. The required area is the shaded region in the figure.

Equilateral triangle bounded by three squares


The Quadrilateral

Quadrilateral is a polygon of four sides and four vertices. It is also called tetragon and quadrangle. In the triangle, the sum of the interior angles is 180°; for quadrilaterals the sum of the interior angles is always equal to 360°

$A + B + C + D = 360^\circ$


Classifications of Quadrilaterals
There are two broad classifications of quadrilaterals; simple and complex. The sides of simple quadrilaterals do not cross each other while two sides of complex quadrilaterals cross each other.

Simple quadrilaterals are further classified into two: convex and concave. Convex if none of the sides pass through the quadrilateral when prolonged while concave if the prolongation of any one side will pass inside the quadrilateral.

21 - 24 Solved problems in maxima and minima

Problem 21
Find the rectangle of maximum perimeter inscribed in a given circle.

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