Planimetry
( plane geometry) - shapes on plane
Point and line
basic geometrical shapes
A not belong to a (
); but B belong to a (
);
Two different lines isn’t crossing each other(parallels || ), or crossing in only one point
*parallels in plane geometry
Line segment AB :
Ray:
*AB & AC are additional to each other
Two line segments are equal if they have the same length:
Angle
Angle (
0,ab,AOB ):
Two angles are equal if they have an equal measure of angle(degree):
Types of angles
ab - acute angle
ac - right angle
ad - obtuse angle
ae - straight angle
*ab & ac - adjacent angles to each other
Vertical angles:
*ab & b1a1 - equal and vertical
a & b - perpendicular lines because c = 90º
Bisector of angle:
ac = cb and c is bisector of ab
Parallel lines
a || b when 1 = 2 (or3 = 4)
*1+3=180º & 2+4=180º
** if two lines parallel to third, then these two lines parallel to each other also
Triangle
Three points which not in one line is triangle ( △ ABC ):
Types of triangles
Equal triangles
Triangles are equal, when:
When three edges are equal in both triangles
When one edge and two nearest angles are equal in both
When two edges and the angle between are equal in both triangles
Similarity of triangles
Triangles are similar, if:
*sum of angles in triangle = 180º
Middle line
AB = c / 2
Area formula
S = ½*h*base
h - perpendicular to base from contrface vertex
Pifagor’s formula
hypotenuse*hypotenuse = leg1*leg1 + leg2*leg2
Tan, Sin, Cos
tan Y = b / a
sin Y = b / c
cos Y = a / c
sin (90 - Y) = cos Y
cos(90 - Y) = sin Y
sinY*sinY + cosY*cosY = 1
1+tgY*tgY = 1/(cosY*cosY)
1+1/(tgY*tgY) = 1/(sinY*sinY)
Cartesian coordinate system
Middle point
x(c) = (x1+x2)/2
y(c) = (y1+y2)/2
*Thales theorem
Distance between points
AB = sqrt( (y2-y1)^2 + (x2-x1)^2 )
Shape equations
Line: ax+by+c = 0
Circle: (x-a)^2+(y-b)^2=r^2
Transformations
Symmetry
To point:
To line:
Homothety
OA1 = k*OA
OB1 = k*OB
OC1 = k*OC
Parallel transfer
x2 = x1 + a
y2 = y1 + b
Vectors
Vector is a line segment with direction
*two lines have one direction if can be equal after parallel transfer
Coordinates of vectors: ( (x2-x1),(y2-y1 )
Operations with vectors
Summary:
va(a1,a2)+vb(b1,b2) =vc((a1+b1),(a2+b2))
Subtraction:
va(a1,a2)-vb(b1,b2) =vc((a1-b1),(a2-b2))
Multiply to number:
va(a1,a2)*k=va(k*a1,k*a2)
Scalar product:
va*vb = |va|*|vb|*cos(ab)
Circle
Shape built from points with equal distance from a defined point(center of the circle)
Inscribed circle(with triangle)
Circumscribed circle(with triangle)
Angles in circle
ABC = AOC/2 :
If AC = A1C1 then ABC = A1B1C1 :
S=pi*r^2
L=2*pi*r
*L(for specific angle) = 2*angle(rad)*r
Quadrangles
Shapes built by 4 points and where no any 3 points in one line
*and edges of this shape are not crossing
Parallelogram
(AB || CD) & (AD || BC)
S = h * base
Rectangle
A = B = C = D = 90º
S = a * b
Rhomb
a || c & b || d & a = b = c = d
S = ½(small diagonal * large diagonal)
Square
(a || c) & (b || d) & (a = b = c = d) & (A = B = C = D = 90º)
S = a * a
Trapezoid
a || b
AB - middle line; AB = (a+b)/2
S = (a+b)*h/2
