Introduction to geometry word problems

Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially the body of the practical knowledge is concerning with the lengths, areas, and volumes. (Source:wikipedia)

In this article we shall discuss about geometry word problems.

Geometry Word Problems Involving Area:

A rectangle is 4 times as extended as it is wide. If the measurement lengthwise is more than 4 inches and the width is less than 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?

Solution:

Let x = original width of rectangle

Area of the rectangle is

A = l w

Plug in the values from the question

60 = (4x + 4) (x –1)

Use distributive property to remove brackets

60 = 4x2 – 4x + 4x – 4

Put in quadratic formula 4x2 – 4 – 60 = 0

4x2 – 64 = 0

This quadratic written as a difference of two squares

(2x) 2 – (8)2 = 0

Factorize Difference of two squares are

(2x) 2 – (8)2 = 0

(2x – 8)(2x + 8) = 0

Therefore the values for x is

Since x is a dimension, it would be positive.

So, we take x = 4

The question wants the dimensions of the original rectangle.

The width of the given rectangle is 4.

The length is 4 times the width = 4 × 4 = 16

Answer: The dimensions of the rectangle are 4 and 16.

Geometry Word Problems Involving Perimeter:

 A triangle has a perimeter of 50. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?

Solution:

Let x = length of the equal side

The formula for perimeter of rectangle

P = sum of the three sides

Plug in the values from the question

50 = x + x + x+ 5

Combine like terms

50 = 3x + 5

Isolate variable x

3x = 50 – 5

3x = 45

x =15

The question requires length of the third side.

The length of third side is = 15 + 5 =20

Answer: The length of third side is 20

Practice Geometry Word Problems:

A triangle has a perimeter of 80. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?
                   Answer: The length of third side is 30

A triangle has a perimeter of 95. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?
                   Answer: The length of third side is 35

Various types of conics sections

Definition of conic sections: We can define conic section as follows: Consider a double cone. If we have a plane that cuts this double cone, the cross section thus obtained at the intersection of the plane and the cone is called a conic section.

Depending on the angle at which the plane cuts the cone and the position of the plane, we can have mainly 4 different conics: circle, ellipse, parabola and hyperbola.

Conic s have been studied for over 2000 years. Greek mathematician Apollonius studied them intensely. He wrote a book ‘the conic’ that remained a standard work on the topic for eighteen centuries. In the sixteenth century, Galileo declared that the trajectory of a projectile was a parabola. The reflectors in head lights of a car, the speakers in a sound system and the mirrors in a telescope are all in the shape of a parabola.

Parabolic mirrors are used to harness solar energy. In the seventeenth century, Kepler declared that planets revolved around the sun in elliptical orbits. It is because of our knowledge of ellipses that precise predictions of time and place of solar and lunar eclipses is possible. The path of motion of a comet in the solar system is also in the shape of a parabola, ellipse or a hyperbola. Knowledge of conics is extremely useful in such terrestrial sciences as architecture and bridge building.

Thus, the study of conics has been proved very useful in space sciences, mechanics, optics, engineering, architecture and other fields.

How to graph conic sections?
To be able to graph conics, let us look at the following concept. Suppose line l is a fixed vertical line and another line m intersects l in the point V and makes an angle of measure a(0 < a < pi/2) with it. If m is made to rotate around V in such a way that a remains constant, then the surface generated is called a double cone. The point V is the vertex and the line m is a generator of the double cone. The line l is the axis of the double cone.  Then the plane cutting this double cone defines the four conics as follows: (see picture below)

Solving Solid Geometry

In solid geometry we study three dimensional geometry (3-D geometry).
For examples: Cube, cuboid, cylinder, cone, sphere, Pyramids, Prisms etc. Dimensions are the terms as length, width, height, thickness etc. A three dimensional figure must have length, width and height.

Cube : A three dimensional shape having equal length(a), width(a) and height (a).
Cuboid: A three dimensional shape having different length(l), width(w) and height(h).
Cylinder: A three dimensional shape having two circular faces of radius(r) at two ends and a curved surface of height(h).
Cone: A three dimensional shape having a circular face at one end and a curved surface of height(h).
Sphere: A three dimensional shape of radius r. For example: a ball.

Formulas for Solving Solid Geometry
(1) Cube :           Lateral Surface Area ( Area of four sides i.e. front, back, left, right ) = 4a2
                           Total Surface Area( Area of all six faces) = 6a2
                            Volume = a x a x a = a3
(2) Cuboid:        Lateral Surface Area ( Area of four sides i.e. front, back, left, right ) = 2h(l+w)
                            Total Surface Area( Area of all six faces) = 2( lw + wh + hl )
                             Volume = lwh
(3) Cylinder:      Curved surface area = 2Ï€rh
                            Total surface area (including two circles on both ends) = 2Ï€r(r+h)
                            Volume = 2Ï€r2h
(4) Cone :          Curved surface area = Ï€rl where l is the slant height of the cone l = `sqrt(h^2 + r^2)`
                            Total surface area (including a circles on the base) = Ï€r(r+l)
                            Volume = 1/3 Ï€r2h
(5) Sphere :      Surface area = 4Ï€r2
                            Volume = 4/3 Ï€r3

How to Solve Problems for Solid Geometry
Step 1) Make a figure of solid given in the problem.
Step 2) Write the dimensions of the solid e.g. length, width, height, radius etc.
Step 3) Apply the formula for particular solid geometry figure.
Step 4) Write the unit of the particular physical quantity e.g. square meters, cubic centimeters etc.

Circles and semi circles

Important definitions related to circles:

1. Circle: A circle is a simple closed curve all of whose points are at a constant distance from a fixed point in the same plane. The fixed point is called the centre of the circle.

2. Circumference of a circle: The distance right around the circle is called its circumference. It is the perimeter of the circle. The traditional method to measure this perimeter of a circle was using a thread or a rope long the circumference. However this method is not too practical for very large circular fields or pieces of land. Therefore for all practical purposes, the following formula was derived by mathematicians for circumference of a circle.
 C = pi*d. Where, C = circumference of the circle, d = diameter of the circle and pi = ratio of the circumference of a circle to its diameter. The value of the Greek letter pi (read as pi) was experimentally calculated by mathematicians. It is an irrational number. A decimal of nonrecurring type. It is a constant. Indian mathematician Ramanujan gave two approximations for the value of pi in the year 1914. Generally all mathematicians have accepted the value of this constant as 3.141 592 653 589 793....

3. Semi circle: A diameter divides a circle into two equal parts which are called semi circles. The length of the curved portion of a semi circle is equal to half the circumference of the circle. The total perimeter of a semi circle is equal to sum of half the circumference and diameter. So putting that mathematically, perimeter of a semi circle = P,
P = C/2 + d, where C = circumference of a circle with diameter d.
From any point on the semi circle if we draw two lines that meet both the ends of the diameter, the angle so formed is called angle in a semi circle. This angle in a semi circle is always a right angle.

4. Unit of a circle (or unit circle): A circle which has a unit radius is called a unit circle. In other words a circle with radius = 1 and diameter = 2 is called a unit circle.

5. Intersecting circles: If there are two circles in a plane then any of the following three possibilities are there:
(a) The circles do not touch or intersect each other at all.
(b) The circles touch each other in exactly one point.
(c)  The circles intersect each other in exactly two points.

Beginners Guide to Geometry of Circle

A branch of Mathematics, Geometry is a study of the size, shape and position of two and three dimensional figures. Geometry of a Circle is a study of a circle, its parts and its properties. A math circle is an important and special figure and as such its parts have special names. Circle in Geometry is a planar figure in which all points are equidistant from a fixed point. This fixed point is called the centre of the circle. A segment with one endpoint at the centre of the circle and the other endpoint on the curve of the circle is a radius; the plural of radius is radii.

A segment whose endpoints lie on the circle is called the chord. A chord that passes through the centre of the circle is called the diameter of the circle. Let us learn more about Geometry circle, there are special lines and line segments in a circle; like secant, tangent and point of tangency.  Any line that contains a chord is called a secant. A tangent of a circle is a line in the plane of the circle that intersects the circle in exactly one point. The point where the tangent intersects a circle is called the point of tangency. Let us now learn a bit about tangents of a circle, a line that is tangent to two circles in the same plane is a common tangent. A common tangent that intersects the segment joining the centers of two circles is an external common tangent.

Now that we have a brief introduction to a circle and its parts, let us learn about the geometry circle formula. The major formulas in circles are as given below:
• Diameter is twice the radius. d=2r
• Circumference of a circle is the distance around the outer edge. It is like the perimeter of a circle. It is calculated using the formula 2 pi r, where r is the radius and pi is taken as 3.14
• Area of a circle is given by pi r2, r is the radius and pi value is taken as 3.14
• An arc is a part of circle. Length of an arc can be calculated using the formula, (a/360)x 2 pi r
• A sector is a portion of a circle bounded by two radii and the arc joining the radii. Area of a sector in degrees is, (sector angle/360) pi r2 and in radians it is (sector angle/2) r2

Know more about the solid geometry, Math Homework Help. This article gives basic information about geometry circle. Next article will cover more Geometryconcept and its advantages,problems and many more. Please share your comments.

Different kinds of graph

Lets learn kinds of graph below. We have learn t what is a bar graph a few days back.

Look below for the kinds of graph

Kind 1: Pictograph.

Kind 2: Bar graph.

Kind 3: Line graph.

Kind 4: Scatter plot.

These are some important graphs in mathematics, we will learn

Absolute values - Inequation and complex number

The absolute value of an integer is the numerical value of  the integer regardless of its sign .the absolute value of any integer say , a is denoted by |a|.On the number line the absolute value of an integer is regarded as the distance from  a point irrespective of its sign. The absolute value of a integer is always positive .
Some Absolute value examples are |-5| = 15 , |13|= 13.

Complex number

Absolute value inequality
To understand absolute value inequality , we will take few examples.

Example1 : |3x| ≤ 6
To  solve absolute inequality , here  we will use the absolute inequality results
|x|≤ a  =>  -a ≤  x ≤  a

=>  |3x|≤ 6
=>  -6 ≤  3x ≤ 6
=> Divide both sides by 3, we have
=>  -2 ≤  x ≤  2

Absolute value of complex number.

If a , b are two real number , then a number  a+ ib is  called as complex number.

Real and imaginary part of complex number : if z = a+ib  is a complex number , then a is called the real part of z and b is known as the imaginary part  of z . The real part of z is denoted by Re(z) and imaginary part is denoted by Im(Z).

Complex Number

The plane in which we represent a  complex number  geometrically  is known  as complex plane or argand plane or the Gaussian plane The point Z plotted on the argand plane is called the argand diagram.The length of the line segmemt 0z is called the absolute  of  complex number z and is denoted by |z|.thus |z|=√x²+y ²

= √(Re(z))²+(Im(z))²  In the above given figure z = 3+ 3i , so absolute of z , |z| = √3²+3 ²

=√18 = 3√2

Absolute value equations and inequalities

Now let us solve absolute value equations

Example1: Solve absolute value equation|x+ 5|= 4

Solution : For solving absolute value equations we will consider two cases

=> x + 5 = 4 or x + 5  = -4

=> x= -1 or x =-9 ans

Example 2: Solve the absolute inequality |x-2| ≥ 5

Solution : For solving absolute inequality , we will use the result

|x-a|≥ r => x ≤ a-r  or x ≥ a+r

=> |x-2| ≥ 5 => x ≤ 2-5  or x ≥ 2+5

=> X ≤ 3  or x ≥ 7

=> X ∈ ( -∞ , -3] or x ∈ [ 7 ,∞)

The solution set  of absolute inequality  is ( -∞ , -3] ∪ [ 7 ,∞)