Vaša košarica je prazna
Appendix
GEOMETRY
The position of lines
e
f
M
one common point
e
f
no common points
(or an infinite number of common points)
(or an infinite number of common points)
e
f
no common points
intersecting
lines
parallel
lines
lines
skew
coplanar
not coplanar
perpendicular lines • two lines are perpendicular if they divide the plane into four congruent parts; symbol: e ⏊ f; on a drawing ⦜
angle • two rays originating from the same point divide the plane into two angles; the rays are the arms of the angle, and the starting point is the vertex
types of angles
- If the arms of an angle are perpendicular, the smaller angle formed by them is called a right angle. Symbol: ⦜
arm
angle
arm
P
vertex
- If the two arms form a straight line, the angle formed by them is called a straight angle.
- If the two arms are on top of each other, the larger angle formed by them is called a full angle, while the smaller one is called a zero angle.
Acute angles are larger than zero angles but smaller than right angles. Obtuse angles are larger than right angles but smaller than straight angles. Reflex angles are larger than straight angles but smaller than full angles. Reflex angles are concave, while the rest are convex.
zero angle < acute angle < right angle < obtuse angle < straight angle < reflex angle < full angle
concave plane figure • a plane figure is concave if there is a line segment connecting two points of the figure that does not entirely lie inside the figure
convex plane figure • a plane figure is convex if every point on any line segment connecting two points of the figure lies inside the figure
concave
concave solid • a solid is concave if there is a line segment connecting two points of the solid that does not entirely lie inside the solid
convex solid • a solid is convex if every point on any line segment connecting two points of the solid lies within the solid
convex
The circle
circle • a set of points in the plane found at the same distance from a given point
disc • a plane figure bounded by a circle
radius • a line segment connecting the centre and any point of the circle
circumference
disc
diameter(d)
radius
0
(r)
diameter • a line segment passing through the centre and connecting two points of the circle (symbol: \latex{ d}); the length of the diameter is two times the length of the radius \latex{(d = 2 \times r)}
arc • a segment of the circle
sector • the area between two radii and their connecting arc
arc
sector
0
sphere • a solid bounded by curved surfaces; it is a set of points found at the same distance from a given point in space; the given point is the centre \latex{ (O) } of the sphere, while the distance is its radius \latex{ (r) }; the word sphere may refer to the spherical surface and the solid itself
radius of the sphere • a line segment connecting the centre and any point of the spherical surface
diameter of the sphere • a line segment passing through the centre that connects two points of the spherical surface; symbol: \latex{d}; the diameter of the sphere is twice the length of the radius \latex{(d = 2 \times r)}
centre (0)
diameter (d)
radius (r)
square • a square is a rectangle whose sides are equal in length
perimeter of the square • four times the length of its side; the perimeter of the square with sides \latex{ a } is:
\latex{P = 4 \times a}area of the square • the length of the sides multiplied by the same number; the area of a square with sides \latex{ a } is:
\latex{A = a \times a}
A
D
B
C
a
a
a
a
\latex{ a }
\latex{ a }
\latex{A=a\times a}
rectangle • a rectangle is a quadrilateral whose adjacent sides are perpendicular to each other
perimeter of a rectangle • two times the sum of the lengths of two adjacent sides; the perimeter of a rectangle with sides a and b is:
\latex{P = 2 \times a + 2 \times b = 2 \times (a + b)}area of a rectangle • the product of the lengths of two adjacent sides; the area of a rectangle with sides a and b is:
\latex{A = a \times b}
A
D
B
C
\latex{ b }
\latex{ a }
\latex{A=a\times a}
perimeter of a polygon • the sum of the lengths of the sides
Cuboids
surface of a cube • the surface area of the cube with \latex{ a } long edges:
\latex{A = 6 \times a\times a}
volume of a cube • the volume of a cube with \latex{ a } long edges:
\latex{V = a \times a \times a}
surface area of a cuboid • the surface area of a cuboid with \latex{ a, b } and \latex{ c } long edges:
\latex{A = 2 \times a \times b + 2 \times a \times c + 2 \times b \times c = 2 \times (a \times b + a \times c + b \times c)}
volume of a cube • the volume of a cube with \latex{ a } long edges:
\latex{V = a \times a \times a}
surface area of a cuboid • the surface area of a cuboid with \latex{ a, b } and \latex{ c } long edges:
\latex{A = 2 \times a \times b + 2 \times a \times c + 2 \times b \times c = 2 \times (a \times b + a \times c + b \times c)}
Cuboids
Square prisms
Cubes
volume of a cuboid • the volume of the cuboid is the product of the length of the edges at a vertex; the volume of the cuboid with \latex{ a, b } and \latex{ c } long edges:
\latex{V = a \times b \times c}
surface area of a solid • the surface area of a solid bordered by plane figures is the sum of the areas of the faces; its symbol is: A
side
edge
vertex
NUMBERS
natural numbers • natural numbers are whole numbers including zero; they are denoted by the symbol ℕ
ℕ = {\latex{ 0; 1; 2; 3; 4; ... }}
positive integers • positive integers are whole numbers greater than zero; symbol: \latex{ + } (plus); example: \latex{ 5 } or \latex{ +5 } (plus \latex{ 5 })\latex{ \gt } the \latex{ + } sign in front of the number is usually left out
Integers
Natural
numbers
numbers
\latex{ 0 }
\latex{ 7 }
\latex{ 2 }
\latex{ 5 }
\latex{ 1 }
\latex{ -4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ -5 }
\latex{ -20 }
negative integers • negative integers are whole numbers less than zero; symbol: \latex{ - } (minus); example: \latex{ -5 } (minus \latex{ 5 })
inverse of a number • numbers that are the same distance from zero on the number line are called inverses of each other; the inverse of zero is zero
absolute value • the absolute value of a number is its distance from zero on a number line; symbol: \latex{|–5|} \latex{ = +5 } (the absolute value of minus \latex{ 5 } is plus \latex{ 5 });
since distance cannot be negative, the absolute value of a number is always positive or \latex{ 0 }
since distance cannot be negative, the absolute value of a number is always positive or \latex{ 0 }
– The absolute value of a positive number is the number itself.
– The absolute value of zero is zero.
– The absolute value of a negative number is the inverse of the number.
– The absolute value of zero is zero.
– The absolute value of a negative number is the inverse of the number.
simple fraction
interpretation of fractions \latex{ 1 }. • when a whole is divided into equal parts, and some of the parts are taken away, you get a fraction
The numerator shows how many of the equal parts are chosen.
The denominator shows how many equal parts \latex{ 1 } whole is divided into.
interpretation of fractions \latex{ 2 }. • a fraction is the quotient of two whole numbers
interpretation of fractions \latex{ 2 }. • a fraction is the quotient of two whole numbers
division sign
dividend
divisor
quotient
\latex{ 3 } divided into
\latex{ 4 } equal parts.
\latex{ 4 } equal parts.
Divide a whole into \latex{ 4 } equal parts and take \latex{ 3 } of them.
fraction
denominator
numerator
÷
\latex{ \frac{3}{4} }
\latex{ 3 }
\latex{ 4 }
Average
average of two numbers • the sum of two numbers divided by two
average of three numbers • the sum of three numbers divided by three
average of \latex{ a } and
\latex{ b }:
\latex{ b }:
\latex{(a+b)\div2}
frequency of data • it refers to the number of times a certain piece of information occurs
average of \latex{ a, b }
and \latex{ c }:
and \latex{ c }:
\latex{(a+b+c)\div3}
