June 2012

Common Quadrilaterals

Square

 
square.gif

 

Area, $A = a^2$

Perimeter, $P = 4a$

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

 

Quadrilateral Circumscribing a Circle

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

Tangential Quadrilateral

 

Area,

$A = rs$

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

Derivation for area

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.
 

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.
 

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?
 

030-projections-of-d.gif

 

Example 01 - Quadratic equation problem

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?
 

Example 02 - Quadratic equation problem

Problem
Determine the equation whose roots are the reciprocals of the roots of the equation 3x2 - 13x - 10 = 0.
 

Example 03 - Sum and product of roots of quadratic equation

Problem
Find the sum and product of roots of the quadratic equation x2 - 2x + 5 = 0.
 

Example 01 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Solve for x, y, and z from the following simultaneous equations.
 

$z^x \, z^y = 100\,000$   ←   equation (1)

$(z^x)^y = 100\,000$   ←   equation (2)

$\dfrac{z^x}{z^y} = 10$   ←   equation (3)
 

System of Equations

System of Linear Equations

The number of equations should be at least the number of unknowns in order to solve the variables. System of linear equations can be solved by several methods, the most common are the following,

1. Method of substitution
2. Elimination method
3. Cramer's rule
 

Many of the scientific calculators allowed in board examinations and class room exams are capable of solving system of linear equations of up to three unknowns.
 

Example 02 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Find the value of x, y, and z from the following equations.
$xy = -3$   →   Equation (1)

$yz = 12$   →   Equation (2)

$xz = -4$   →   Equation (3)
 

Example 03 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Find the value of x, y, and z from the given system of equations.
$x(x + y + z) = -36$   →   Equation (1)

$y(x + y + z) = 27$   →   Equation (2)

$z(x + y + z) = 90$   →   Equation (3)
 

Example 04 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Solve for x, y, and z from the following system of equations.
$x(y + z) = 12$   →   Equation (1)

$y(x + z) = 6$   →   Equation (2)

$z(x + y) = 10$   →   Equation (3)
 

Arithmetic, geometric, and harmonic progressions

Elements
a1 = value of the first term
am = value of any term after the first term but before the last term
an = value of the last term
n = total number of terms
m = mth term after the first but before nth
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum of the 1st n terms
 

Smallest number for given remainders

Problem
Find the smallest number which when divided by 2 the remainder is 1, when divided by 3 the remainder is 2, when divided by 4 the remainder is 3, when divided by 5 the remainder is 4, and when divided by 6 the remainder is 5.