# Distance between projection points on the legs of right triangle (solution by Calculus)

**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?

# Length of one side for maximum area of trapezoid (solution by Calculus)

**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.

# Length of one side for maximum area of trapezoid (solution by Geometry)

**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

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**

# Common Quadrilaterals

**Square**

Perimeter,

Diagonal,

# 01 Minimum distance between projection points on the legs of right triangle

**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?

# 09 Dimensions of smaller equilateral triangle inside the circle

**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.

# 01 Area enclosed by line rays inside a square

**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.

# 10 Area common to three squares inside the regular hexagon

**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.