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N.CHANDRA SEKHAR
SA (MATHS)
Z.P.HIGH SCHOOL
MADANAPALLE
MADANAPALLE MANDAL
CHITTOOR DISTRICT
A.P
POLY GEO BOARD
VERIFICATION INSTRUMENT OF
PROPERTIES,THEOREMS AND
PROBLEMS OF GEOMETRICAL
FIGURES
SOUTH INDIA SCIENCE FAIR - 2016
Visvesvaraya Industrial & Technological Museum(VITM)
BANGALORE
th
From 19 January to 23rd January of 2016
TITLE: POLY GEO BOARD
Name of the School
Name of the Teacher
Mobile Number
Name of the District
Name of the State
Mail Id
: Z.P.High School
MADANAPALLE
Madanapalle Mandal
Chittoor District
A.P
: N.CHANDRA SEKHAR
: 09440650217
: Chittoor
: Andhra Pradesh
: nchandrasekhar563@gmail
BRIEF DESCRIPTION ABOUT THE EXHIBIT
“VERIFICATION INSTRUMENT OF PROPERTIES, THEOREMS AND PROBLEMS OF
GEOMETRICAL FIGURES”
Need of the Exhibit / Introduction:
It is verification Instrument to verify the Angles (i.e. Acute angles, Obtuse angles, Right
angle, Straight angle, Reflexive angle, Corresponding angles, Alternate angles, Vertically opposite
angles, Adjacent angles, Opposite angles).
To verify the properties of Triangles(i.e. Based on angles – Right angle triangle, Acute
angle triangle, Obtuse angle triangle. Based on sides – Equilateral triangle, Isosceles triangle, Scalene
triangle).
To verify the properties of Quadrilaterals(i.e. Trapezium, Parallelogram, Rectangle,
Rhombus, Square).
To verify the Basic proportionality theorem, Pythagoras Theorem, Mid point theorem,
Alternate segment theorem ets.
Objective:
1. Child centered maths tool
2. It is purely designed for easy learning /understanding the concepts of Geometrical
figures.
3. Low expensive, Rough use, no need of electricity/battery.
4. 6th to 10th class teaching aid.
Required materials:
Iron frames, rubber bands, pro circles, metal scale.
Procedure:
Welding the iron frames with perfect measurement and attach to wooden board. Grain
the screws, put this screws in between the frames and attach the pro circles to that screws, to move
the screws on frames using rubber bands we make different Geometrical figures and measure angles,
lengths and verify the properties of Triangles, Quadrilaterals.
Uses:
After completion of each concept related to this poly geo board the children learn by
own and verify the concepts through examples .
In Architecture
Carpentry and people that make furniture
Engineering department
Clocks/ watches
Ship navigation
Sports – Throwing Javelin
Bridges
Roof of our houses
Ramps
Especially in art,design and architecture
To create floor plans for new buildings
Signature of the Teacher
ANGLES
1. Acute angle: It is more than 0 degrees and below 90 degree
2. Right angle:
It is 90 degrees only
3.Obtuse angle: It is more than 90 degrees and below 180 degrees
4.Straight angle:
It is 180 degrees only
5.Reflexive angle: It is more than 180 degrees below 360 degrees
6.Complete angle:
It is 360 degrees only
Triangles
7.Equilateral triangle: All sides of a triangle are equal.
8.Isosceles triangle: Any two sides of a triangle are equal
9.Scalene triangle: Any two sides of a triangle are not equal
10.Acute angled triangle: All angles are below 90 degrees
11.Right angled triangle:
Only one angle is 90 degrees
12.Obtuse angled triangle: Only one angle is more than 90 degrees
13. Transversal : A transversal is a line which passes through two
lines at two distinct points.
Interior angles
: C,D,E,F
Exterior angles
: A,B,G,H
Corresponding angles : 4 pairs ( If given lines are parallel )
A = E , B = F, C = G, D = H
Alternate angles
: 2 pairs ( If given lines are parallel )
C = F,D = E.
One side of transversal interior angles are supplementary ( If given lines are parallel )
14.
Sum of the angles in a triangle is 180 degrees
15.Theorem: In a triangle equal angles opposite sides are equal
16.Theorem: In a triangle external angle is equal to the sum of
interior opposite angles
17. Quadrilateral: A closed figure having four sides.
Property
: Sum of all angles are 360 degrees.
18.Trapezium:
Property :
One pair of opposite sides are parallel.
Property of Quadrilateral.
Non parallel opposite sides adjacent angles are supplementary.
19.Parallelogram:
Two pairs of opposite sides are parallel.
Properties: Properties of Trapezium.
Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.
Each diagonal divides the parallelogram into two congruent triangles.
Any two adjacent angles have their sum is equal to 180.
20.Rectangle :
Properties :
A parallelogram in which one angle is right angle
is called Rectangle.
Properties of parallelogram.
Each angle is 90 degrees.
Diagonals are equal.
21.Rhombus : A parallelogram in which two adjacent sides are
equal is called Rhombus.
Properties:
Properties of Parallelogram.
All sides are equal.
Diagonals are perpendicularly bisect each other.
22. Square: A rectangle in which adjacent sides are equal is called a
square. (OR)
A rhombus in which one of its angle is a right angle is
called a square.
Properties: Properties of Rectangle and Rhombus.
Each diagonal divides it into two congruent right angle isosceles
triangles.
23. Theorem : Parallelograms on the same base and between the
same parallels are equal in area.
n
Area of ABCD = Area of ABGH
24.Theorem : Triangles on the same base and between the same
parallels are equal in area.
Area of ABC = Area of PBC
25.Theorem: If a parallelogram and a triangle are on the same base
and between the same parallels then the area of
triangle is half the area of the parallelogram.
Area of ABE = ½ Area of ABCD
Problem: Show that the median of a triangle divides it into two
triangles of equal areas.
26. Theorem: If two lines intersect each other, then the pairs of
vertically opposite angles thus formed are equal.
27. S.A.S Congruence rule: Two triangles are congruent if two sides
and included angle of one triangle are
equal to the two sides and the included
angle of the other triangle.
28. A.S.A Congruence rule: Two triangles are congruent ,if two
angles and the included side of one
triangle are equal to two angles and the
included side of the other triangle.
29. S.S.S Congruence rule: In two triangles, if the three sides of one
triangle are respectively equal to the
corresponding three sides of another
triangle, three the two triangles are congruent.
30.R.H.S Congruence rule: If in two right angle triangles the
hypotenuse and one side of one triangle are
equal to the hypotenuse and one side of the
another triangle, then the two triangles are
congruent.
31. Theorem : Angles opposite to equal sides of an isosceles triangle
are equal.
32. Theorem: If two sides of a triangle are unequal, the angle
opposite to the longer side is larger.
33.Theorem: The sum of any two sides of a triangle is greater the
third side.
34.Corollary: Show that the angle bisectors of a parallelogram form
a rectangle.
35.Theorem: The line segment joining the mid points of two sides of
a triangle is parallel to the third side and also half of it.
36. Problem : In triangle ABC S,Q and R are the mid points of sides AB,BC
and CA respectively. Show that triangle ABC is divided into four congruent
triangles, when the three midpoints are joined to each other.
37.Basic Proportionality Theorem( Thales Theorem ): If a line is
drawn parallel to one side of a triangle to intersect the other two
sides in distinct points, then the other two sides are divided in the
same ratio.
AD/DB = AE/ED
38.Theorem: If a perpendicular is drawn from the vertex of the right
angle of a right triangle to the hypotenuse, then the triangles on
both sides of the perpendicular are similar to the whole triangle and
to each other.
39. Pythagoras Theorem: In a right angle triangle, the square of
hypotenuse is equal to the sum of the
squares of the other two sides.
40.Problem: Prove that the area of a triangle formed by joining the
middle points of the sides of a triangle is equal to one fourth area of
the given triangle.
41. Problem: Prove that the area of a quadrilateral formed by
joining the middle points of consecutive sides of a quadrilateral is
equal to half the area of the given quadrilateral.
42.Theorem: If two triangles have a common vertex and their bases
are on the same straight line, the ratio between their areas is equal
to the ration between the lengths of their bases.
Area of ABM/ Area of AMC = BM/MC
43.Theorem: The areas of similar triangles are proportional to the
squares on corresponding sides.
Area of ABC/Area of DEF = AB²/PQ² = BC²/QR² = AC²/PR²
44. Corollary : The ratio of the areas of two similar triangles is equal
to the ratio of the squares on any two corresponding altitude.
Area of ABC/Area of PQR = AM²/PN²
45. Corollary : The ratio of the areas of two similar triangles is equal
to the ratio of the squares on any two corresponding medians.
46. Corollary : The ratio of the areas of two similar triangles is equal
to the ratio of the squares on their perimeters.
Area of ABC/Area of DEF = (AB+BC+CA)²/(DE+EF+DF)²
47.Theorem: The locus of the point which is equidistance from two
fixed points is the perpendicular bisector of the line segment joining
the two fixed points.
48.Theorem: The locus of a point which is equidistance from
intersecting straight lines consists of a pair of straight line which
bisect the angles between the two given lines.
49.Circle: It is a closed plane curve such that each point on it is at a
constant distance from a fixed point laying on the same plane.
50. Circumcircle: The circle passing through the vertices of a triangle
is called circumcircle.
51.Cyclic Quadrilateral: The circle passing through the vertices of a
quadrilateral is called Cyclic quadrilateral.
52.Theorem:Thy pairs of opposite angles of a cyclic quadrilateral are
supplenentry.
53.Theorem: The exterior angle of a cyclic quadrilateral is equal to
the interior opposite angle.
54.Theorem:
An angle in a semicircle is a right angle.
55.Theorem: Angles in the same segment of a circle are equal.
56.Theorem: Angle subtended by an arc at the centre of circle is
twice the angle subtended by it at any other point on the circle.
57.Theorem: Equal chords of a circle subtend equal angles at the
centre.
58.Theorem : Equal chords are at equal distance from the centre of
the circle.
59.Theorem: Arcs of equal length subtend equal angles at the
centre.
60. Theorem: The tangent at any point of a circle is perpendicular to
the radius through the point of contact.
61.Theorem: The lengths of tagents drawn from an external point to
a circle are equal.
62.Theorem: The straight line drawn from the centre of a circle to
the midpoint of a chord (which passes through the centre ) is at
right angles to the chord.
63.Theorem: There can be one and only one circle passing through
three non collinear points .
64.Theorem: If two chords are equal,they cut off equal arcs.
65.Theorem: If two chords of a circle intersect internally or
externally the the product of the lengths of their segments are
equal.
66.Theorem: If a chord and tangent intersect externally then the
product of the lengths of the segments of the chord is equal to the
square of the lengths of the tangent from the point of contact to the
point of intersection.
67.Alternate segment Theorem: If a line touches the circle from the
point of contact a chord is drawn, the angles between the tangent
and the chord are respectively equal to the angles in the
corresponding alternate segments.
68. Symmetry: A geometrical figure is said to have line symmetry or axis
symmetry if it can be divided into two identical halves by a line drawn
through it. The line dividing the figure into two congruent parts , i.e two
symmetric halves, is known as the line of symmetry.
A figure is said to have point symmetry about a point O if every
line segment joining two points on the boundary of the figure and also
passing through O gets bisected at O. The point O is known as the point of
symmetry or centre of symmetry.
69. Pentagon:
70.Hexagon:
71.Heptagon:
72.Octogon:
73.Nonagon:
74.Decagon:
75.Slope of a line: If angle is given
HF
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