# Circumference of a Circle - cvut.cz

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Circumference of a Circle

The line segment AB, |AB| = 2r , and its interior point X are given. The sum of the lengths of semicircles over the diameters AX and XB is

3πr ; πr ; 3

Circumscribed circle

Circumcircle of a polygon is a circle which passes through all the vertices of the polygon.

Inscribed circle

Incircle of a polygon is a circle touches (is tangent to) all sides of the polygon.

Sarka Voracova Plane Geometry May 30, 2018 2 / 26

Circle Inscribed in Hexagon

The ratio of the area of a regular hexagon with side a to the area of the circle inscribed in it is

2 √

Inscribed triangle

Consider an equilateral triangle ABC with the side length a and an inscribed equilateral triangle DEF , where D ∈ AB, E ∈ BC , F ∈ CA and the area of the triangle DEF is equal to one third of the area of the triangle ABC . The length of the side of the triangle DEF is

a√ 2

Sarka Voracova Plane Geometry May 30, 2018 4 / 26

Consider an equilateral triangle ABC with the side length a and an inscribed equilateral triangle DEF , where D ∈ AB, E ∈ BC , F ∈ CA and the area of the triangle DEF is equal to one half of the area of the triangle ABC . The length of the side of the triangle DEF is

a

2 ;

Cartesian Coordinate System in the Plane

Any point P in the plane can be located by unique ordered pair on numbers – coordinates. We write P[x , y ] or P = (x , y).

Example

Describe and sketch the regions given by following sets

{(x , y); x ≥ 0} {(x , y); y = 1} {(x , y); y < 1}

Sarka Voracova Plane Geometry May 30, 2018 6 / 26

Square in CCS

Determine coordinates of vetices C ,D of the square ABCD, where A[1,−1],B[5, 1]1

1square Sarka Voracova Plane Geometry May 30, 2018 7 / 26

Lines – Slope form of the equation

Slope m – measure of the steepness. The slope of line AB is

m = y

x .

Let line l passes through a given point A = (xA, yA) and has slope m.

X = (x , y) ∈ l ⇔ y − yA x − xA

= m

Lines – Slope and general equation

Example

Write the equation of the line l = AB, where A = (2, 1) and B = (0, 3).

Determine:

The slope m of the line l .

Intersection points with coordinate axes, P = l ∩ x , Q = l ∩ y .

whether the line l goes through the point P = (3, 4).

Sarka Voracova Plane Geometry May 30, 2018 9 / 26

Identical Straight Lines2

Straight lines 6x + by + c = 0 and AB, where A[1, 2], B[2, −1], are identical only if

b = 4 ∧ c = 10

b = 4 ∧ c = −14

b = 2 ∧ c = −10

b = −4 ∧ c = 2

b = −2 ∧ c = 10

2IdenticalLines.ggb Sarka Voracova Plane Geometry May 30, 2018 10 / 26

Lines – Parametric equations

Directional vector of line AB is any vector parallel with ~u = B − A. Directional vector ~u is perpendicular to normal vector ~n, i.e. ~u · ~n = 0. Let line l passes through a given point A = (xA, yA) and has directional vector ~u.

X = (x , y) ∈ l ⇔ ~AX = t · ~u X − A = t · ~u

X = A + t · ~u

Point of Intersection3

A straight line x + 2y − 7 = 0 and a line segment x = 1 + 4t, y = 1 + 2t, t ∈ 0, 1,

do not intersect

intersect at point [4, 3 2 ]

3IntersectionPoint.ggb Sarka Voracova Plane Geometry May 30, 2018 12 / 26

Perpendicular Bisector4

A straight line ax − 2y + c = 0 is the axis of the line segment AB, where A[1, 5], B[−3, 3], only if

a = −4 ∧ c = 4

a = 1 ∧ c = 3

a = 4 ∧ c = −4

a = 2 ∧ c = −2

a = 7 ∧ c = 0

4PerpendicularBisector Sarka Voracova Plane Geometry May 30, 2018 13 / 26

Reflection in Line5

Given a line a and a point P. Reflection P ′ of P in a is the point such that PP ′ is perpendicular to a, and PM = MP ′, where M is the point of intersection of PP ′ and a. In other words, P ′ is located on the other side of axis, but at the same distance from a as P. P ′ is said to be a mirror of P.

5ReflectionLineChB.ggb Sarka Voracova Plane Geometry May 30, 2018 14 / 26

Reflection in Line6

Axial symmetry with an axis p : x − y = 0 maps a point A[3, 0] to a point A[?, ?]

6ReflectionLine.ggb Sarka Voracova Plane Geometry May 30, 2018 15 / 26

Reflection in Line

Axial symmetry with an axis p : x − 2y + 1 = 0 maps a point A[4, 0] to a point

[0, 4]

[0, 3]

[3, 2]

[2, −2]

[2, 4]

Altitude of a Triangle7

A straight line containing the height ha of the triangle ABC , where A[0, −1], B[6, 0], C [4, 3], has an equation

7AltitudeTriangle.ggb Sarka Voracova Plane Geometry May 30, 2018 17 / 26

Altitude of a Triangle

A straight line containing the height hc of the triangle ABC , where A[−1, 1], B[3, 2], C [2, 5], has an equation

4x − y − 3 = 0

x + 4y − 22 = 0

x − 4y + 18 = 0

x + 4y + 13 = 0

4x + y − 13 = 0

Equation of the Circle

Circle is the set of all points X in a plane that are at a given distance r from a given point, the centre O. X [x , y ]; O[m, n]

r = |OX | = |(X − O)| = √

(x −m)2 + (x − n)2

Sarka Voracova Plane Geometry May 30, 2018 19 / 26

Circumcircle and Incircle8

Write the equation for circle inscribed(circumscribed) in a square ABCD, where A[1,−1], B[2, 2]

8InscribedSquare.ggb Sarka Voracova Plane Geometry May 30, 2018 20 / 26

Circumcircle about Rectangle

The equation of a circle circumscribed about the rectangle ABCD, where A[2, −3], C [8, 3], is

(x − 3)2 + (y − 3)2 = 36

x2 + 10x + y2 − 18 = 0

x2 − 10x + y2 + 7 = 0

(x − 10)2 + y2 − 72 = 0

x2 − 10x + y2 + 28 = 0

Sarka Voracova Plane Geometry May 30, 2018 21 / 26

Inscribed Circle

The equation of a circle inscribed in the square ABCD, where A[2, 1], C [4, 11], is

x2 + y2 − 6x − 12y + 32 = 0

x2 + y2 − 3x − 6y − 29 = 0

x2 + y2 + 6x + 12y − 32 = 0

x2 + y2 + 6x − 12y + 32 = 0

x2 + y2 − 6x + 12y + 29 = 0

Sarka Voracova Plane Geometry May 30, 2018 22 / 26

Circumcircle about Triangle

Write the equation of the circle c circumscribed about the triangle ABC , where A[0, 0], B[2, 0], C [0, 2]

Sarka Voracova Plane Geometry May 30, 2018 23 / 26

The equation of a circle circumscribed about the triangle ABC , where A[1, 5], B[9, 1], C [1, 1], is

x2 + y2 − 10x − 6y + 14 = 0

x2 + y2 − 5x − 3y + 20 = 0

x2 + y2 − 10x + 6y − 20 = 0

x2 + y2 + 10x + 6y − 54 = 0

x2 + y2 + 5x + 3y + 20 = 0

Sarka Voracova Plane Geometry May 30, 2018 24 / 26

Nearest and Furthest point

Consider a circle given by the equation x2 + y2 − 16x − 12y + 75 = 0 . The ratio of distances of the nearest and furthest points of this circle from the origin of the coordinate system is

1 : 11 1 : 3 2 : 3 1 : 2 3 : 13

Sarka Voracova Plane Geometry May 30, 2018 25 / 26

Circle and Tangent Line9

Consider a circle with the center S [−1, 3] and a tangent t given by the equation x − 2y + 2 = 0 . The equation of this circle is

x2 + y2 + 2x − 6y + 5 = 0

The line segment AB, |AB| = 2r , and its interior point X are given. The sum of the lengths of semicircles over the diameters AX and XB is

3πr ; πr ; 3

Circumscribed circle

Circumcircle of a polygon is a circle which passes through all the vertices of the polygon.

Inscribed circle

Incircle of a polygon is a circle touches (is tangent to) all sides of the polygon.

Sarka Voracova Plane Geometry May 30, 2018 2 / 26

Circle Inscribed in Hexagon

The ratio of the area of a regular hexagon with side a to the area of the circle inscribed in it is

2 √

Inscribed triangle

Consider an equilateral triangle ABC with the side length a and an inscribed equilateral triangle DEF , where D ∈ AB, E ∈ BC , F ∈ CA and the area of the triangle DEF is equal to one third of the area of the triangle ABC . The length of the side of the triangle DEF is

a√ 2

Sarka Voracova Plane Geometry May 30, 2018 4 / 26

Consider an equilateral triangle ABC with the side length a and an inscribed equilateral triangle DEF , where D ∈ AB, E ∈ BC , F ∈ CA and the area of the triangle DEF is equal to one half of the area of the triangle ABC . The length of the side of the triangle DEF is

a

2 ;

Cartesian Coordinate System in the Plane

Any point P in the plane can be located by unique ordered pair on numbers – coordinates. We write P[x , y ] or P = (x , y).

Example

Describe and sketch the regions given by following sets

{(x , y); x ≥ 0} {(x , y); y = 1} {(x , y); y < 1}

Sarka Voracova Plane Geometry May 30, 2018 6 / 26

Square in CCS

Determine coordinates of vetices C ,D of the square ABCD, where A[1,−1],B[5, 1]1

1square Sarka Voracova Plane Geometry May 30, 2018 7 / 26

Lines – Slope form of the equation

Slope m – measure of the steepness. The slope of line AB is

m = y

x .

Let line l passes through a given point A = (xA, yA) and has slope m.

X = (x , y) ∈ l ⇔ y − yA x − xA

= m

Lines – Slope and general equation

Example

Write the equation of the line l = AB, where A = (2, 1) and B = (0, 3).

Determine:

The slope m of the line l .

Intersection points with coordinate axes, P = l ∩ x , Q = l ∩ y .

whether the line l goes through the point P = (3, 4).

Sarka Voracova Plane Geometry May 30, 2018 9 / 26

Identical Straight Lines2

Straight lines 6x + by + c = 0 and AB, where A[1, 2], B[2, −1], are identical only if

b = 4 ∧ c = 10

b = 4 ∧ c = −14

b = 2 ∧ c = −10

b = −4 ∧ c = 2

b = −2 ∧ c = 10

2IdenticalLines.ggb Sarka Voracova Plane Geometry May 30, 2018 10 / 26

Lines – Parametric equations

Directional vector of line AB is any vector parallel with ~u = B − A. Directional vector ~u is perpendicular to normal vector ~n, i.e. ~u · ~n = 0. Let line l passes through a given point A = (xA, yA) and has directional vector ~u.

X = (x , y) ∈ l ⇔ ~AX = t · ~u X − A = t · ~u

X = A + t · ~u

Point of Intersection3

A straight line x + 2y − 7 = 0 and a line segment x = 1 + 4t, y = 1 + 2t, t ∈ 0, 1,

do not intersect

intersect at point [4, 3 2 ]

3IntersectionPoint.ggb Sarka Voracova Plane Geometry May 30, 2018 12 / 26

Perpendicular Bisector4

A straight line ax − 2y + c = 0 is the axis of the line segment AB, where A[1, 5], B[−3, 3], only if

a = −4 ∧ c = 4

a = 1 ∧ c = 3

a = 4 ∧ c = −4

a = 2 ∧ c = −2

a = 7 ∧ c = 0

4PerpendicularBisector Sarka Voracova Plane Geometry May 30, 2018 13 / 26

Reflection in Line5

Given a line a and a point P. Reflection P ′ of P in a is the point such that PP ′ is perpendicular to a, and PM = MP ′, where M is the point of intersection of PP ′ and a. In other words, P ′ is located on the other side of axis, but at the same distance from a as P. P ′ is said to be a mirror of P.

5ReflectionLineChB.ggb Sarka Voracova Plane Geometry May 30, 2018 14 / 26

Reflection in Line6

Axial symmetry with an axis p : x − y = 0 maps a point A[3, 0] to a point A[?, ?]

6ReflectionLine.ggb Sarka Voracova Plane Geometry May 30, 2018 15 / 26

Reflection in Line

Axial symmetry with an axis p : x − 2y + 1 = 0 maps a point A[4, 0] to a point

[0, 4]

[0, 3]

[3, 2]

[2, −2]

[2, 4]

Altitude of a Triangle7

A straight line containing the height ha of the triangle ABC , where A[0, −1], B[6, 0], C [4, 3], has an equation

7AltitudeTriangle.ggb Sarka Voracova Plane Geometry May 30, 2018 17 / 26

Altitude of a Triangle

A straight line containing the height hc of the triangle ABC , where A[−1, 1], B[3, 2], C [2, 5], has an equation

4x − y − 3 = 0

x + 4y − 22 = 0

x − 4y + 18 = 0

x + 4y + 13 = 0

4x + y − 13 = 0

Equation of the Circle

Circle is the set of all points X in a plane that are at a given distance r from a given point, the centre O. X [x , y ]; O[m, n]

r = |OX | = |(X − O)| = √

(x −m)2 + (x − n)2

Sarka Voracova Plane Geometry May 30, 2018 19 / 26

Circumcircle and Incircle8

Write the equation for circle inscribed(circumscribed) in a square ABCD, where A[1,−1], B[2, 2]

8InscribedSquare.ggb Sarka Voracova Plane Geometry May 30, 2018 20 / 26

Circumcircle about Rectangle

The equation of a circle circumscribed about the rectangle ABCD, where A[2, −3], C [8, 3], is

(x − 3)2 + (y − 3)2 = 36

x2 + 10x + y2 − 18 = 0

x2 − 10x + y2 + 7 = 0

(x − 10)2 + y2 − 72 = 0

x2 − 10x + y2 + 28 = 0

Sarka Voracova Plane Geometry May 30, 2018 21 / 26

Inscribed Circle

The equation of a circle inscribed in the square ABCD, where A[2, 1], C [4, 11], is

x2 + y2 − 6x − 12y + 32 = 0

x2 + y2 − 3x − 6y − 29 = 0

x2 + y2 + 6x + 12y − 32 = 0

x2 + y2 + 6x − 12y + 32 = 0

x2 + y2 − 6x + 12y + 29 = 0

Sarka Voracova Plane Geometry May 30, 2018 22 / 26

Circumcircle about Triangle

Write the equation of the circle c circumscribed about the triangle ABC , where A[0, 0], B[2, 0], C [0, 2]

Sarka Voracova Plane Geometry May 30, 2018 23 / 26

The equation of a circle circumscribed about the triangle ABC , where A[1, 5], B[9, 1], C [1, 1], is

x2 + y2 − 10x − 6y + 14 = 0

x2 + y2 − 5x − 3y + 20 = 0

x2 + y2 − 10x + 6y − 20 = 0

x2 + y2 + 10x + 6y − 54 = 0

x2 + y2 + 5x + 3y + 20 = 0

Sarka Voracova Plane Geometry May 30, 2018 24 / 26

Nearest and Furthest point

Consider a circle given by the equation x2 + y2 − 16x − 12y + 75 = 0 . The ratio of distances of the nearest and furthest points of this circle from the origin of the coordinate system is

1 : 11 1 : 3 2 : 3 1 : 2 3 : 13

Sarka Voracova Plane Geometry May 30, 2018 25 / 26

Circle and Tangent Line9

Consider a circle with the center S [−1, 3] and a tangent t given by the equation x − 2y + 2 = 0 . The equation of this circle is

x2 + y2 + 2x − 6y + 5 = 0