Geometry Flashcards
acute angle
острый угол меньше 90 градусов
right angle
прямой угол
obtuse angle
тупой угол 90< x< 180
straight angle
угол 180 градусов
supplementary angle
смежный угол, sum of two angles is 180
complementary angle
if the sum of two angles is 90
polygon
многоугольник
triangle
треугольник
quadrilateral
четырехугольник
pentagon
пятиугольник
hexagon
шестиугольник
sum of interior angle measure of polygon
180*(n-2)
altitude
высота в треугольнике
Triangle. Supplementary angle is
is equal to the sum of the other angles
Triangle. The longest side
The longest side is opposite the largest angle; the shortest is opposite the smallest angle
Triangle.
The sum length of any two sides is greater than the length of third side
Triangle. Area
S=1/2 Base X altitude
scalene triangle
having no sides equal
isosceles triangle
равнобедренный
equilateral triangle
раносторонний
sides of the isosceles triangle sometimes called
legs
Equliateral Triangle. Properties
- All sides are equal
- Angels = 60
- The intersection point of the altitudes is the centre circumscribed curcle about triangle and the center inscribed circle in triangle
- R: r = 2: 1 (соотношение радиусов вписанной и описанной окружностей)
- Area = sqrroot(3)/4 X a^2
Right Triangle.Pythogorean triple
Pythogorean triple: 3 : 4 : 5 5 : 12: 13 8 : 15:17 7:24:25
Right Triangle. 30-60-90 right triangle, the leg sides
C- гипотенуза
1/2С - против угла в 30 градусов
корень из (3)/2 = против угла в 60 градусов
Right Triangle. 45-45-90 right triangle, the leg sides
C- гипотенуза
корень из (2)/2 - оставшиеся стороны
Similar Triangles.
When Corresponding angles of two triangles are equal the triangles are said to be similar.
Angles are similar provided the following three conditions are satisfied:
1. all angles are equal
2. AB/A1B1 = AC/A1C1=BC/B1C1
3.AB/A1B1=AC/A1C1, angel A = A1
Equal Triangles
Two triangles are equal provided the following three conditions are satisfied:
- AB=A1B1, AC=A1C1, BC=B1C1
- AB = A1B1, AC=A1C1, angel A=A1
- AB=A1B1, angle A=A1, angel B=B1
Parallelogram
Two Pairs of parallel sides Properties: 1. The opposite sides are parallel: AB||CD, AD || BC 2. The opposite sides are equal: AB=CD, AD=BC 3. The opposite angles are equal: angel A=C, B=D 4. The diagonals bisect each other: AE=EC, BE=ED 5. The are of a parallelogram is equal to a side S=A *H (высота к основанию) 6. сумма consecutive angles= 180
Rectangle.
Прямоугольник. Это параллелограммам у которого углы 90 градусов.
Properties of a rectangle:
1. the diagonals are equal:
AC=BD
2. The are of a rectangle is equal to the product of length and width: S=a X b
Rhombus
A parallelogram having four equal sides
Properties of a rhombus:
1. The diagonals are perpendicular:
AC parallel BD
2. The diagonals bisect the angles they join:
AC bisects angle A and C, BD bisects angle B and D
3. The are of rhombus is equal to one-half a product of the two diagonals:
S = 1/2 X d1 x d2
or S=a x h
Square
parallelogram having four equal angles and four equal sides
1. The diagonals are equal: AC = BD
2. Diagonals are perpendicular : AC perpendicular BD
3. The diagonals bisect the angles they join
4. The are of a square is equal to its side squared or equal to one-half its diagonal squared:
S=a^2=1/2*d^2
Trapezoid
a qudraliteral having one pair of parallel sides is called trapezoid. The parallel sides are called bases, nonparallel sides called legs.
Area:
S = 1/2 * h (b1+b2)
Circle.
Circumference = 2piR,
Area = piR^2
The length of the arc = 2piR * X/360 (X - угол)
Area of the sector = piR^2*X/360 (X - угол)
pi=3.14
Circle. Angles
Central angel- an angel formed by two radii of a circle is called a central angle
Inscribed angle - formed by two chords having common endpoint on the circumference of a circle
If inscribed angle and central angle intercept the same arc, the inscribed angle will measure exactly half the central angle.
If a triangle is inscribed in a circle so that on of its sides is a diameter of the circle. then the triangle is a right triangle
Circle. Inscribed and curcumscrubed
if each vertex of a polygon lies on a circle. then the polygon is inscribed.(вписанный)
Circubscrbied (описанный)
Hexagon
Polygon having six sides.
The sum of internal angles = 720
Properties:
1. Area S = 6 * sqroot(3)/4 * a^2
2. If R is a radius of the circumscribed circle then R=a
3. if r is a radius of inscribed circle then R=sqroot(3)/2*a
Rectangular solid
Three dimensional figure formed by 6 rectangular surfaces. Each recantgular surface is a face.
6 faces, 12 edges, 8 vertices and 4 diagonals.
V=a*b*c Surfaces area: S = 2 (a*b +b*c + a*c) The length of a diagonal d=sqroot(a^2+b^2+c^2)
Cube
all edges are equal
V=a^3
S=6a^2
d=asqroot(3)
Cylinder
V=pir^2h
total surfaces area:
2piRh (surface area of the side of a cylinder) + 2piR^2
Coordinate plane
I quadrant справа вверху
II quadrant слева вверху
iii quadrant слева вниз
iv quadrant справа внизу
Distance between two points
B(X2,y2) A(x1,y1,)
D=sqroot( (x2-x1)^2 + (y2-y1)^2) )
Midpoint
Xo=(X1+X2)/2
Yo=(Y1+Y2)/2
Slope
y=kx+b
k-slope
slope = (Y2-Y1)/(X2-X1)
Two lines L1,L2 perpendicular, k1*k2= -1, k1,k2 - slopes
The value of the slope gives the angle between the line and x-axis.
Parabola
y=ax^2+bx+c - curve
parabola’s axis of symmetry: x=-b/2a
if a>0 parabola opens upward
if a<0 parabola opens downward
Ромб - признаки
Если в параллелограмме диагонали взаимно перпендикулярны, то этот параллелограмм — ромб.
Если в параллелограмме диагонали являются биссектрисами углов, то этот параллелограмм — ромб.
