Geometrical locus. Circle and circumferenceGeometrical locus. Mid-perpendicular. Angle bisector.
Circumference. Circle. Arc. Secant. Chord. Diameter. Tangent line.
Segment of a circle. Sector of a circle. Angles in a circle.
Central angle. Inscribed angle. Circumscribed angle.
Radian measure of angles. Round angle. Ratio of
circumference length and diameter. Length of an arc.
Huygens' formula. Relations between elements of a circle.
Geometrical locus ( or simply locus) is a totality of all points, satisfying the certain given conditions.
E x a m p l e 1. A midperpendicular of any segment is a locus, i.e. a totality of all points, equally
removed from the bounds of the segment. Suppose that PO AB and AO = OB :
Then, distances from any point P, lying on the midperpendicular PO, to bounds A and B of the segment AB are both equal to d . So, each point of a midperpendicular has the following property: it is removed from the bounds of the segment at equal distances.
E x a m p l e 2. An angle bisector is a locus, that is a totality of all points, equally removed from the angle sides.
E x a m p l e 3. A circumference is a locus, that is a totality of all points ( one of them - A ),
equally removed from its center O.
Circumference is a geometrical locus in a plane, that is a totality of all points, equally removed from its center. Each of the equal segments, joining the center with any point of a circumference is called a radius and signed as r or R . A part of a plane inside of a circumference, is called a circle. A part of a circumference ( for instance, AmB, Fig.39 ) is called an arc of a circle.The straight line PQ, going through two points M and N of a circumference, is called a secant ( or transversal ). Its segment MN, lying inside of the circumference, is called a chord.
A chord, going through a center of a circle ( for instance, BC, Fig.39 ), is called a diameter and signed as d or D . A diameter is the greatest chord of a circle and equal to two radii ( d = 2 r ).
Tangent. Assume, that the secant PQ ( Fig.40 ) is going through points K and M of a circumference. Assume also, that point M is moving along the circumference, approaching the point K. Then the secant PQ will change its position, rotating around the point K. As approaching the point M to the point K, the secant PQ tends to some limit position AB. The straight line AB is called a tangent line or simply a tangent to the circumference in the point K. The point K is called a point of tangency. A tangent line and a circumference have only one common point -- a point of tangency.
Properties of tangent.
1) A tangent to a circumference is perpendicular to a radius, drawing to a point of
tangency ( AB OK, Fig.40 ) .
2) From a point, lying outside a circle, it can be drawn two tangents to the same
circumference; their segments lengths are equal ( Fig.41 ).
Segment of a circle is a part of a circle, bounded by the arc ACB and the corresponding chord AB ( Fig.42 ). A length of the perpendicular CD, drawn from a midpoint of the chord AB until intersecting with the arc ACB, is called a height of a circle segment. Sector of a circle is a part of a circle, bounded by the arc AmB and two radii OA and OB, drawn to the ends of the arc ( Fig.43 ).
Angles in a circle. A central angle -- an angle, formed by two radii of the circle ( AOB, Fig.43 ). An inscribed angle -- an angle, formed by two chords AB and AC, drawn from one common point ( BAC, Fig.44 ).
A circumscribed angle -- an angle, formed by two tangents AB and AC, drawn from one common point ( BAC, Fig.41 ).
A length of arc of a circle is proportional to its radius r and the corresponding central angle :
l = r
So, if we know an arc length l and a radius r, then the value of the corresponding central angle can be determined as their ratio:
= l / r .
This formula is a base for definition of a radian measureof angles. So, if l = r, then = 1, and we say, that an angle is equal to 1 radian ( it is designed as = 1 rad ). Thus, we have the following definition of a radian measure unit: A radian is a central angle ( AOB, Fig.43 ), whose arc's length is equal to its radius ( AmB = AO, Fig.43 ). So, a radian measure of any angle is a ratio of a length of an arc, drawn by an arbitrary radius and concluded between the sides of this angle, to the radius of the arc. Particularly, according to the formula for a length of an arc, a length of a circumference C can be expressed as:
C = 2r,
where is determined as ratio of C and a diameter of a circle 2r:
= C / 2 r .
is an irrational number; its approximate value is 3.1415926...
On the other hand, 2 is a round angle of a circumference, which in a degree measure is equal to 360 deg. In practice it often occurs, that both radius and angle of a circle are unknown. In this case, an arc length can be calculated by the approximate Huygens' formula:
p 2l + ( 2l -- L ) / 3 ,
where ( according to Fig.42 ): p -- a length of the arc ACB; l -- a length of the chord AC;
L -- a length of the chord AB. If an arc contains not more than 60 deg, a relative error of this formula is less than 0.5%.
Relations between elements of a circle. An inscribed angle ( ABC, Fig.45 ) is equal to a half of the central angle ( AOC, Fig.45 ), based on the same arc AmC. Therefore, all inscribed angles ( Fig.45 ), based on the same arc ( AmC, Fig.45 ), are equal. As a central angle contains the same quantity of degrees, as its arc ( AmC, Fig.45 ), then any inscribed angle is measured by a half of an arc, which is based on( AmC in our case ).
All inscribed angles, based on a semi-circle (APB, AQB, ..., Fig.46 ), are right angles ( Prove this, please ! ). An angle (AOD, Fig.47 ), formed by two chords ( AB and CD ), is measured by a semi-sum of arcs, concluded between its sides:
( AnD + CmB ) / 2 .
An angle (AOD, Fig.48 ), formed by two secants ( AO and OD ), is measured by a semi-difference of arcs, concluded between its sides: ( AnD -- BmC ) / 2 . An angle (DCB, Fig.49 ), formed by a tangent and a chord ( AB and CD ), is measured by a half of an arc, concluded inside of it: CmD / 2 .An angle (BOC, Fig.50 ), formed by a tangent and a secant ( CO and BO ), is measured by a semi-difference of arcs, concluded between its sides: ( BmC -- CnD ) / 2 .
A circumscribed angle (AOC, Fig.50 ), formed by the two tangents, (CO and AO), is measured by a semi-difference of arcs, concluded between its sides: ( ABC -- CDA ) / 2 . Products of segments of chords ( AB and CD, Fig.51 or Fig.52 ), into which they are divided by an intersection point, are equal: AO · BO = CO · DO.
A square of tangent line segment is equal to a product of a secant line segment by the secant line external part ( Fig.50 ): OA2 = OB · OD ( prove, please! ). This property may be considered as a particular case of Fig.52.
A chord ( AB, Fig.53 ), which is perpendicular to a diameter ( CD ), is divided into two in the intersection point O :
AO = OB . ( Try to prove this ! ).
there is no easier way to learn Geometry
It means height
It means slanted surface.
The locus point is the perpendicular bisector of AB. The locus point is the perpendicular bisector of AB.
yeah
what does empirical mean in geometry
If you feel that you have control over your life, you have an internal locus of control. If, on the other hand, you feel that you are at the whims of fate, you have an external locus of control.
geometry means lines, segments, and points!!
Geometry means earth or land measurements
there is no easier way to learn Geometry
No, the word for place in Latin is "locus".
It is an education term that mean it meets the states criteria for geometry.
In geometry, magnitude is the length of the hypotenuse of a right triangle.
Lonely sacred place.
The specific place or point at which someone or something is held responsible
"Equal" means they are the same whether in geometry or another math.
The plural form of locus is loci.