Transcript Document 7195632

Good Afternoon!
Today we will be learning about
Review of Geometry
Let’s warm up :
The dimensions of Rectangular prisms are given.
Find their volume.
1)
Height = 5 cm
Width = 5 cm
Depth = 5 cm
2)
1) 125 cm3
3)
Height = 4 cm
Width = 3 cm
Depth = 6 cm
2) 72 cm3
Height = 8 cm
Width = 10 cm
Depth = 9 cm
4)
3) 720 cm3
Height = 7 cm
Width = 7 cm
Depth = 7 cm
4) 343 cm3
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Points, lines, segments, rays
Geometry is all about shapes and their properties.
The two most common subjects in geometry are:
1) Plane Geometry
2) Solid Geometry
Plane geometry: is the study of plane figures in the
plane such as points, lines, line segments, rays, angles,
circles, triangles, quadrilaterals, and other polygons ...
shapes that can be drawn on a piece of paper.
Solid Geometry: is the study of three dimensional
objects like cubes and pyramids. It is called threedimensional, or 3D because there are three
dimensions: width, depth and height.
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Point
X
A point is a location in space.
 A point is an exact location .
 Points are dimensionless,
i.e., a point has no width, length, or height.
We locate points relative to some arbitrary standard
point, often called the "origin".
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Line
DE
D
E
A line is a group of points on a straight path
that extends to infinity.
Any two points on the line can be used to name it.
This line is called line DE.
 Its length, having no limit, is infinite.
It has no width or height.
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Line segment
X
XY
Y
A line segment is a part of a line that has two end points.
A line segment is the path of shortest distance between
two points.
The two end points of the line segment are used to name
the line segment. This line segment is called segment XY.
All the points "between" the two points make up a line segment.
A line segment has one dimension, length. It has
no width or height.
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Ray
O
OP
P
A ray is part of a line. A ray extends indefinitely in one
direction, but ends at a single point in the other
direction. That point is called the end-point of the ray.
A ray is named starting with its end point first and
then any other point on the ray second.
This ray is called ray OP.
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Using the graphic figure:
T
X
U
Y
V
W
1) Name a line.
Z
1) VY
2) Name a line segment with U as an end point.
3) Name a ray with V as an end point.
3) VY
4) Name a line segment with X as an end point.
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2) UT
4) XU
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Perpendicular, parallel & intersecting lines
Lines are parallel if they are always the same distance
apart (called "equidistant"), and will never meet.
l
m
Lines m and l are parallel lines.
They will travel to infinity in either direction
and never intersect.
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Intersecting lines
Two or more lines that meet at a point are
called intersecting lines. That point would be
on each of these lines.
x
Q
y
In the Figure, lines x and y are intersecting
lines and intersect at point Q.
Lines can only intersect at one point and only one point.
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Perpendicular lines.
If the line segments meet or cross each other to form
square corners, they are perpendicular to each other.
s
right angles
t
The little box drawn in the corner, means "at right angles“.
Perpendicular lines intersect at a point and form 4 right angles.
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Symbols in Geometry
Here are the some geometrical symbols:
Symbol
Meaning
||
parallel
perpendi
cular
Right
angle
Example
In Words
EF||GH
The line EF is parallel to
line GH
AB CD
The line AB is
perpendicular to line CD
is 90°
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A right angle is 90
degrees
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Now you try!
Classify each pair of lines as parallel, intersecting, or perpendicular.
1)
2)
2) intersecting
1) parallel
3)
4)
4) parallel
3) perpendicular
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Angles (right, acute, obtuse) protractor
What Is an Angle?
An angle is a combination of two rays with a common endpoint.
B
angle AOB
vertex
O
A
arm
The endpoint (O) is known as the vertex of the angle
And
the rays (OA and OB) are called the sides or arms of the angle .
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Angles On a Straight Line
If we know one angle is 45°, what is angle “x" ?
x
45°
Angle x will be 180° − 45° = 135°
This method can be used to find angles on one side
of a straight line.
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Angles Around a Point
Angles around a point will always add up to 360 degrees.
110°
40°
150°
60°
The angles here all add to 360°.
40° + 110° + 150° + 60° = 360°
Because of this, if there is an unknown angle we can always find it.
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Complementary Angles
Two Angles are Complementary if they add up to
90 degrees (a Right Angle).
60°
30°
These two angles (40° and 50°) are Complementary Angles,
because they add up to 90°.
But the angles don't have to be together to
Complement each other.
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Supplementary Angles
Two Angles are Supplementary if they add up to
180 degrees (a Straight Angle).
60°
120°
These two angles (120° and 60°) are Supplementary
Angles, because they add up to 180°.
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Now you try!
Find the Complement of the following:
1
?
1) 57°
2
59°
?
33°
2) 31°
Find the Supplement of the following:
3) 60°
3
?
4) 45°
4
120°
?
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135°
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Triangles (isosceles, equilateral, right)
A triangle is one of the basic shapes of geometry:
A polygon with three corners or vertices
and three sides or edges which are line segments.
A
a
vertex
B
c
b
C
arm
angle ACB
Or
angle c
The three angles always add to 180°.
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Interior Angle: An Interior Angle is
an angle inside a shape.
Exterior Angle: The Exterior Angle is the angle between
any side of a shape, and a line extended from the next side.
Exterior Angle
Interior Angle
45°
135°
If you add up the Interior Angle and Exterior Angle
you get a straight line, 180°.
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Triangle Classification
The basic elements of any triangle are its sides and vertices.
Triangles are classified depending on relative sizes of their elements.
Triangles can be classified according to their internal angles.
Acute Triangle: An acute triangle is a triangle
whose angles are all acute (i.e. less than 90°).
In the acute triangle shown above, a, b and c are all acute angles.
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Right Triangle: A right triangle is a triangle
with a right angle (i.e. 90°).
The side opposite the right angle is always the triangle's longest side.
It is called the hypotenuse of the triangle.
The other two sides are called the legs.
leg
a°
hypotenuse
90°
right angle
b°
leg
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Obtuse Triangle: An obtuse triangle has one
obtuse angle (i.e. greater than 90º).
The longest side is always opposite the obtuse angle.
a°
>90°
b°
In the obtuse triangle shown above, a is the obtuse angle.
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Types of Triangles
There are three special names given to triangles
that tell how many sides (or angles) are equal.
The triangle classification is summarized as follows:
Equilateral Triangle: An equilateral triangle has all
three sides equal in length. Its three angles are also
equal and they are each 60º.
a
a
60º
60º
60º
a
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Isosceles Triangle: An isosceles triangle has
two sides of equal length. The angles opposite
the equal sides are also equal.
a
a
xº
xº
Scalene Triangle: A scalene triangle has no sides of
equal length. Its angles are also all different in size.
b
xº
a
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zº
yº
c
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Now you try!
Classify each triangle as Equilateral, Isosceles or Scalene :
1)
2cm
2cm
2)
4 cm
4 cm
2cm
2) Isosceles
1) Equilateral
Classify each triangle as Acute, Right or Obtuse :
3)
4)
95 º
3) Obtuse
4) Right
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Quadrilaterals and other polygons (rectangle,
square, rhombus, parallelogram, trapezoid)
A polygon is a plane shape with straight sides.
But the sides have to be straight, and it has to be 2-dimensional.
A quadrilateral is a 4-sided polygon, just like a
triangle is a 3-sided polygon, a pentagon is a 5sided polygon, and so on.
There are many different kinds of quadrilaterals, but all have
several things in common: all of them have four sides, are
coplanar, have two diagonals, and the sum of their four interior
angles equals 360 degrees.
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Types of Quadrilaterals
The Square: A Square is a four-sided shape
which has all the sides equal and where every
angle is a right angle (i.e. 90°).
Also opposite sides of a square are parallel.
A square also fits the definition of a rectangle (all angles
are 90°), and a rhombus (all sides are equal length).
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The Parallelogram: Opposite sides are parallel and
equal in length, and opposite angles are equal (angles
"a" are the same, and angles "b" are the same).
a
b
a
b
NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!
The Trapezoid (or Trapezium): A trapezoid has
one pair of opposite sides parallel.
A trapezoid is not a parallelogram because only
one pair of sides is parallel.
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Classify each quadrilaterals as rectangle,
square, rhombus, parallelogram, trapezoid:
4)
1) rectangle
6)
5)
2) trapezoid
7)
3) square
CONFIDENTIAL
4) parallelogram
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Congruence
Two polygons are congruent if they are the same
size and shape that is, if their corresponding
angles and sides are equal.
If one shape can become another using Turns, Flips and/or
Slides, then the two shapes are called Congruent:
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Congruent Angles
Congruent Angles have the same angle in degrees.
135°
The angles don't have to point in the same direction.
They don't have to be on similar sized lines.
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Congruence of triangles
A triangle has three sides and three angles.
If two triangles are congruent, then the sides and
angles that match are called corresponding parts.
A
B
C
Let's look at the corresponding parts of triangles ABC and DFE.
 Angle A corresponds to angle D.
 Angle B corresponds to angle F.
 Angle C corresponds to angle E.
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A
B
C
 Side AB corresponds to side DF.
 Side BC corresponds to side FE.
 Side CA corresponds to side ED.
Congruent figures are named in the order of their corresponding
parts. Here, we say "triangle ABC is congruent to triangle DFE,"
because vertex A corresponds to vertex D, vertex B corresponds
to vertex F, and vertex C corresponds to vertex E.
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Now you try!
Write whether these figures are congruent.
1)
2)
1) congruent
3)
2) Not congruent
4)
4) congruent
3) Not congruent
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BREAK
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GAME
Click on the link below for some exciting puzzle
http://www.thekidzpage.com/onlinejigsawpuzzles/kidsjigsaw-puzzles/12-piece-jigsaw/03-27-07littlebrother.html
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Reflections, rotations and translations
If one shape can become another using Turns, Flips
and/or Slides, then the two shapes are called Congruent:
The three main Transformations are:
 Reflection : Flip!
 Rotation : Turn!
 Translation : Slide!
After any of those transformations (turn, flip or slide), the
shape still has the same size, area, angles and line lengths.
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Reflection
A reflection over a line, is a transformation in which each
point of the original figure (pre-image) has an image that is
the same distance from the line of reflection as the original
point but is on the opposite side.
The central line is called the Mirror Line, and it doesn't
matter what direction the mirror line goes, the reflected
image is always the same size, it just faces the other way.
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Rotation
When we "rotate" an object round a point.
We can notice that
The distance from the center to any point on the shape stays the same!
and
Every point makes a circle around the center.!
"Rotation" means turning around a center.
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Rotation
A rotation is a transformation, that moves every point
around a fixed point (usually the origin).
A rotation creates a figure that is congruent to the
original figure and preserves distance and orientation .
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Translation
In Geometry, "Translation" simply means Moving ..
without rotating, resizing or anything else, just moving.
Every point of the shape must move:
* the same distance
* in the same direction.
A translation is a transformation that slides every point
of a figure the same distance in the same direction.
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Now you try!
Write Reflection, Rotation or Translation to
describe how the figure was moved:
1)
2)
1) Translation
2) Reflection
3)
3) Rotation
4)
4) Rotation
5)
5) Translation, Reflection
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Similarity and Symmetry
Similar: Two shapes are Similar if the only difference is size.
If one shape can become another using Resizing,
then the shapes are Similar.
Example:
When two shapes are similar, then:
 corresponding angles are equal, and
 the lines are in proportion.
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Sometimes it can be hard to see if two shapes are
Similar, because you may need to turn, flip or
slide one shape as well as resizing it.
Resized
Resized and
Reflected
Resized and
Rotated
These shapes are all Similar.
If one shape can become another using Resizing,
then the shapes are Similar.
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Fold this picture in half. The two parts
match exactly. This picture has “symmetry.”
Line of symmetry
Symmetry: When a picture or figure has symmetry, it can be
folded in half so that the two parts match exactly.
Where you fold the shape, or the fold line, is called
the line of symmetry.
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Line Symmetry
A figure has line symmetry if it can be folded in half so
that the two halves match exactly i.e. one half
of it is the mirror image of the other half.
Line symmetry is also called bilateral symmetry.
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Figures can have any number of lines of symmetry,
from no lines of symmetry to an infinite, or
unlimited, number of lines of symmetry.
No lines of
symmetry
One line of
symmetry
Two lines of
symmetry
Infinite lines
of symmetry
The Line Symmetry is sometimes called Reflection
Symmetry or Mirror Symmetry.
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Rotational Symmetry
Rotational Symmetry: A figure has rotational symmetry if it
can be rotated about a point less than a full turn to make the
figure look the same as it did before the rotation.
Quarter
turn
Half
turn
3-Quarter
turn
With rotational Symmetry, the shape or image can be rotated
clockwise or counterclockwise 180°and it still looks the same.
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Point Symmetry
Point Symmetry: is when every part has a matching part.
* the same distance from the central point
* but in the opposite direction.
Point Symmetry is sometimes called Origin
Symmetry, because the "Origin" is the central
point about which the shape is symmetrical.
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Now you try!
Write whether of figures are similar or not:
1)
2)
1) similar
2) not similar
Is the dotted line a line of symmetry:
3)
4)
3) Yes
4) No
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Circles and circumference (compass)
Circle: A circle is a shape with all points that are
same distance from the center.
Radius
O
The circle is named circle O since the center is at point O.
Radius: The radius is a line segment that begins
from the centre and touches any point on the circle.
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Diameter: The distance across a circle through the
center is called the diameter.
The Diameter is equal to twice the radius.
Diameter = 2 × Radius
Diameter
O
Circumference: The distance around a circle is
called the circumference.
The circumference of a circle is also called the
perimeter of the circle.
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Lines in a Circle
The name of a line in a circle depends on its position in the circle.
secant
chord
tangent
A secant is a line that passes through any two points on a circle.
A chord is a line that joins two points on the circumference of a circle.
A tangent is a line that touches the circle at only one point.
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Parts of a Circle
An arc is a part of the circumference.
Here, AB is the arc.
O
sector AOB
B
A
arc AB
A sector is the part of a circle between two radii.
Here, AOB is the sector .
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Parts of a Circle
A segment is the part of a circle that is between a chord
and the circumference.
semicircle
segment
A semicircle is a half of a circle.
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Circumference
The Circumference is the distance around the edge of the circle.
It is exactly Pi (the symbol is π) times the Diameter, so:
Circumference = π × Diameter
Since the Diameter is equal to twice the radius.
So this is also true:
Circumference = 2 × π × Radius
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The radius of a circle is 2 inches.
What is the circumference?
The radius of a circle = 2 inches
We know that,
Circumference = 2 × π × Radius
Circumference = 2 x π x 2
= 2 x 3.14 x 2
= 12.56
Replace radius with 2.
Replace π with 3.14.
The circumference of a circle = 12.56 inches
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Now you try!
The radii of the circle are given. Find is the diameter?
1) 44 cm
1) 22 cm
2) 35 ft
2) 70 ft
The radii of the circle are given. Find is the circumference?
3) 23 cm
4) 19 ft
3) 144.44 cm
4) 119.32 ft
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A 2-dimentional figure is a shape with length
and width.
It can be open or closed.
Closed figures: These are those figures that start
and end at the same point.
Open figures: These are those figures that do not
start and end at the same point.
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A polygon is a closed 2-dimentional figures with
straight sides. They are made of straight lines,
and the shape is "closed“.
Polygon
(straight sides)
Not a Polygon
(has a curve)
Not a Polygon
(open, not closed)
A circle is a closed figure, but it does not
have straight sides.
A circle is not a polygon.
A polygon can be grouped by the number of sides they have.
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3-dimensional figure
A 3-dimentional figure: is a figure with length,
width and height.
You can describe a 3- dimensional figure by its parts.
vertex
A face is a flat side.
A base is a face on which the figure sits.
edge
face
base
An edge is where two faces meet.
A vertex is where 3 or more faces meet.
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Examples of 3-dimensional figure.
Cube
Cylinder
A cube has 6 faces,
12 edges, 8 vertices
2 circular bases
Rectangular
pyramid
Triangular pyramid
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Make a Net
You can make a net for any solid figure.
Net for a square pyramid.
Cube
Triangular Prism
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Volume of a Cuboids
Cuboids are a 3-dimensional shape.
It has 3 different measurements.
Height
Width
The volume of he cube is found using
the formula
Volume = Height × Width × Depth
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The dimensions of a Rectangular prism is given.
Find its volume.
Height = 6 cm
Width = 4 cm
Depth = 7 cm
We know that the volume of he cube is found using the formula
Volume = Height × Width × Depth
=6×4×7
= 168
The volume of a Rectangular prism = 168 cm2.
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Identify 3-dimentional figure.
Tell how many faces, edges and vertices it has
1)
2)
1) 4 faces, 6 edges, 4
vertices
3)
3) 1 circular base
2) 6 faces, 12 edges,
8 vertices
4)
4) no face, edge or vertices
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You have done a nice job.
See you in the next session.
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