Problem
In a quadratic equation problem, one student made a mistake in copying the coefficient of x and got roots of 3 and -2. Another student made a mistake in copying the constant term and got the roots of 3 and 2. What are the correct roots?
 

Problem
BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.
 

Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.
 

Area,

Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2
 

Derivation for area

Square

 

 

Area, $A = a^2$

Perimeter, $P = 4a$

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

Problem
From the right triangle ABC shown below, AB = 40 cm and BC = 30 cm. Points E and F are projections of point D from hypotenuse AC to the perpendicular legs AB and BC, respectively. How far is D from AB so that length EF is minimal?
 

Problem
From the figure shown, ABC and DEF are equilateral triangles. Point E is the midpoint of AC and points D and F are on the circle circumscribing ABC. If AB is 12 cm, find DE.
 

Problem
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.
 

Problem
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.