2. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
POINT:
A No dimension.
C
B It’s a position.
Always in CAPITAL letters.
3. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
LINE: It’s an addition of several
points following the same
r
direction.
Always in small letters; r, s, t…
r r
s r
A s
s
Two lines cut each Two lines can be When the two lines
other when they share parallel when the share no point, they
a point. sharing point is in the cross each other.
infinite.
4. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
HALF LINE: One point is known and the
A
∞→ other is in the infinite.
r
A point in the line defines tow
←∞ A ∞→ half-lines, one to the left and
r the other to the right.
SEGMENT:
A B Is a kind of line defined
r between two known points.
5. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
CURVED LINE:
A curved line is a group of
points constantly changing
direction.
Always in small letters.
6. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
PLANE:
Is the set of points that arise when you move a straight
line in one direction.
We need the following information to define a plane:
Non aligned 3 points. Two lines cutting each other.
Two parallel lines. A line and a point out of the line.
7. THEME 2: BASIC PATHS IN THE PLANE
Lines within a plane:
Bisecting line:
8. THEME 2: BASIC PATHS IN THE PLANE
Lines within a plane:
To draw a perpendicular from “M” point outside the line:
9. THEME 2: BASIC PATHS IN THE PLANE
Lines within a plane:
To draw a perpendicular from “P” point inside the line:
10. THEME 2: BASIC PATHS IN THE PLANE
Lines within a plane:
To construct a perpendicular at the end of a given line:
11. THEME 2: BASIC PATHS IN THE PLANE
Lines within a plane:
● To draw parallel lines with the set squares:
12. THEME 2: BASIC PATHS IN THE PLANE
Lines within a plane:
● To draw perpendicular lines with the set squares:
13. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
ANGLES:
Is a measure of a turn. We use a protractor to measure an angle.
Sometimes we use letters from Greek alphabet to name angles; α, β, γ,
δ…
And sometimes we name (B) the vertex of the angle and (choosing A and C
points) on the two sides; we write ABC. So the angle reads ABC.
Different kind of angles:
Null angle: α = 0°
Acute angle: α < 90°
Right angle: α = 90°
Obtuse angle: α > 90°
Plain angle: α = 180°
Complete angle: α = 360°
14. THEME 2: BASIC PATHS IN THE PLANE
Basic geometrics elements:
ANGLES:
Two lines cutting each other at point O creates the following angles;
β
α γ
δ
Adjacent angles: α and β. Same vertex and side in common.
Angles opposite at vertex; α and γ; β and δ.
So, α and γ / β and δ are of the same value.
15. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
To construct an angle similar to a given angle;
16. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
Summing up angles;
17. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
Difference between angles;
18. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
To bisect an angle (bisector);
19. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
To bisect an angle (bisector);
20. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
Drawing angles;
60 angle: 90 angle:
45 angle: 30 angle:
15 angle: 75 angle:
21. THEME 2: BASIC PATHS IN THE PLANE
Operations with angles :
Drawing angles;
105 angle: 120 angle:
135 angle: 150 angle:
22. THEME 2: BASIC PATHS IN THE PLANE
Geometric places:
The set of points having the same geometric characteristics.
1. Circumference:
2. Bisecting line:
23. THEME 2: BASIC PATHS IN THE PLANE
Geometric places:
3. Bisector line:
4. The loci arc of a segment (depending on the angle):
24. THEME 2: BASIC PATHS IN THE PLANE
Circumference:
A circle is a plain figure bounded by a curved line called the
circumference, witch is always equidistant from the centre.
Lines of a circumference:
Radius; Any of the straight lines from the centre to
the circumferences. The radius is half the diameter of
the circumference.
Diameter: The longest possible chord of a
circumference. A line passing through the centre with
both ends touching the circumference.
25. THEME 2: BASIC PATHS IN THE PLANE
Circumference:
Chord: A straight line, witch each end touching the
circumference.
Arrow; It’s a part of the radius between the chord
and the circumference. The radius is perpendicular
to the chord.
Secant: A line that cuts the circumference at two
points.
Tangent: A line touching the circumference at one
point. Forms a right angle with a radius of the circle.
T is the point contact.
27. THEME 2: BASIC PATHS IN THE PLANE
Circumference:
To construct a circumference when you have 3 points.
28. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
TRIANGLES:
Is a polygon formed by three segments.
The addition of every inner angles of a triangle is always 180º.
α + β + γ = 180º
The value of the outside angle of a triangle is the addition of
the two non-adjacent inside angles.
29. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
TRIANGLES:
In every triangle, any side is always smaller than the addition of
the other two;
a<b+c
And any side is larger than the subtraction of the other two;
b>a-c
In every triangle the larger angle is in front of the larger side;
c > a; γ > α
30. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
CLASSIFICATION OF TRIANGLES:
Depending on sides;
Equilateral: Isosceles: Scalene:
31. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
CLASSIFICATION OF TRIANGLES:
Depending on angles;
Acute:
Right:
Obtuse:
32. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
REMARKABLE LINES AND POINTS OF A TRIANGLE:
Bisector / Incentre / Inscribed circle to a triangle.
33. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
REMARKABLE LINES AND POINTS OF A TRIANGLE:
Bisecting line / Circumcentre / Circumscribed circle to a triangle.
34. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
REMARKABLE LINES AND POINTS OF A TRIANGLE:
Altitudes / Orthocentre.
35. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
REMARKABLE LINES AND POINTS OF A TRIANGLE:
Baricentre or Centre of Gravity.
36. THEME 3: TRIANGLES, SQUARES AND REGULAR POLYGONS
CONSTRUCTING TRIANGLES:
a) Knowing the 3
sides a, b and c.
b) Knowing 2 of the
sides and the angle
between them.
c) Knowing one
side, a, and the
angles B and C.
37. THEME 3: TRIANGLES, QUADRILATERALS AND REGULAR POLYGONS
QUADRILATERAL:
It is an polygon formed by 4 sides.
QUADRILATERALS
Trapezium (two
PARALELOGRAM
sides are parallels, Trapezoid (no
(Two by two, sides
the other two parallel sides)
are parallel)
aren’t)
38. THEME 3: TRIANGLES, QUADRILATERALS AND REGULAR POLYGONS
QUADRILATERAL:
Square
Rectangle.
PARALELOGRAM
(Two by two,
sides are parallel)
Rhombus.
Rhomboid
39. THEME 3: TRIANGLES, QUADRILATERALS AND REGULAR POLYGONS
QUADRILATERAL:
Trapezium
(two sides are parallels, the other
two aren’t)
Isosceles
Right
Scalene
41. THEME 3: TRIANGLES, QUADRILATERALS AND REGULAR POLYGONS
REGULAR POLYGONS:
What is a Polygon?
A closed plane figure made up of several line
segments that are joined together.
The sides do not cross each other.
Exactly two sides meet at every vertex.
42. THEME 3: TRIANGLES, QUADRILATERALS AND REGULAR POLYGONS
REGULAR POLYGONS:
One polygon is regular if all the sides and all the angles are equal.
l = Side.
a = Apoteme
r = Radius
α = 180º - (360º / n)
λ = 360º / n