Circumference of a Circle

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1 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 2 πr; 5 4 πr; 1 2 πr; Šárka Voráčová Plane Geometry May 30, / 26

2 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. Šárka Voráčová Plane Geometry May 30, / 26

3 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 3 : π, 3 : π, 2 2 : π, 3 2 : π, 2 : π, Šárka Voráčová Plane Geometry May 30, / 26

4 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 ; a 2 ; a 4 ; a 6 ; a 3 ; Šárka Voráčová Plane Geometry May 30, / 26

5 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 ; a 3 ; a 4 ; a 2 ; a 6 ; Šárka Voráčová Plane Geometry May 30, / 26

6 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} Šárka Voráčová Plane Geometry May 30, / 26

7 Square in CCS Determine coordinates of vetices C, D of the square ABCD, where A[1, 1], B[5, 1] 1 1 square Šárka Voráčová Plane Geometry May 30, / 26

8 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 = (x A, y A ) and has slope m. X = (x, y) l y y A x x A = m y y A = m(x x A ) y = mx x A + y A Šárka Voráčová Plane Geometry May 30, / 26

9 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). Šárka Voráčová Plane Geometry May 30, / 26

10 Identical Straight Lines 2 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 2 IdenticalLines.ggb Šárka Voráčová Plane Geometry May 30, / 26

11 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 = (x A, y A ) and has directional vector u. X = (x, y) l AX = t u X A = t u X = A + t u Šárka Voráčová Plane Geometry May 30, / 26

12 Point of Intersection 3 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 [5, 1] intersect at point [1, 3] intersect at point [3, 2] intersect at point [4, 3 2 ] 3 IntersectionPoint.ggb Šárka Voráčová Plane Geometry May 30, / 26

13 Perpendicular Bisector 4 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 4 PerpendicularBisector Šárka Voráčová Plane Geometry May 30, / 26

14 Reflection in Line5 Given a line a and a point P. Reflection P 0 of P in a is the point such that PP 0 is perpendicular to a, and PM = MP 0, where M is the point of intersection of PP 0 and a. In other words, P 0 is located on the other side of axis, but at the same distance from a as P. P 0 is said to be a mirror of P. S a rka Vora c ova Plane Geometry May 30, / 26

15 Reflection in Line 6 Axial symmetry with an axis p : x y = 0 maps a point A[3, 0] to a point A[?,?] 6 ReflectionLine.ggb Šárka Voráčová Plane Geometry May 30, / 26

16 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] Šárka Voráčová Plane Geometry May 30, / 26

17 Altitude of a Triangle 7 A straight line containing the height h a of the triangle ABC, where A[0, 1], B[6, 0], C[4, 3], has an equation 7 AltitudeTriangle.ggb Šárka Voráčová Plane Geometry May 30, / 26

18 Altitude of a Triangle A straight line containing the height h c 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 Šárka Voráčová Plane Geometry May 30, / 26

19 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 r 2 = (x m) 2 + (y n) 2 Šárka Voráčová Plane Geometry May 30, / 26

20 Circumcircle and Incircle 8 Write the equation for circle inscribed(circumscribed) in a square ABCD, where A[1, 1], B[2, 2] 8 InscribedSquare.ggb Šárka Voráčová Plane Geometry May 30, / 26

21 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 x x + y 2 18 = 0 x 2 10x + y = 0 (x 10) 2 + y 2 72 = 0 x 2 10x + y = 0 Šárka Voráčová Plane Geometry May 30, / 26

22 Inscribed Circle The equation of a circle inscribed in the square ABCD, where A[2, 1], C[4, 11], is x 2 + y 2 6x 12y + 32 = 0 x 2 + y 2 3x 6y 29 = 0 x 2 + y 2 + 6x + 12y 32 = 0 x 2 + y 2 + 6x 12y + 32 = 0 x 2 + y 2 6x + 12y + 29 = 0 Šárka Voráčová Plane Geometry May 30, / 26

23 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] Šárka Voráčová Plane Geometry May 30, / 26

24 The equation of a circle circumscribed about the triangle ABC, where A[1, 5], B[9, 1], C[1, 1], is x 2 + y 2 10x 6y + 14 = 0 x 2 + y 2 5x 3y + 20 = 0 x 2 + y 2 10x + 6y 20 = 0 x 2 + y x + 6y 54 = 0 x 2 + y 2 + 5x + 3y + 20 = 0 Šárka Voráčová Plane Geometry May 30, / 26

The ratio of distances of the nearest and furthest points of this circle

25 Nearest and Furthest point Consider a circle given by the equation x 2 + y 2 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 Šárka Voráčová Plane Geometry May 30, / 26

26 Circle and Tangent Line 9 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 x 2 + y 2 + 2x 6y + 5 = 0 9 CircleTangent.ggb Šárka Voráčová Plane Geometry May 30, / 26

Geometry Period Unit 2 Constructions Review

Name 2-7 Review Geometry Period Unit 2 Constructions Review Date 2-1 Construct an Inscribed Regular Hexagon and Inscribed equilateral triangle. -Measuring radius distance to make arcs. -Properties of equilateral