GEOMETRY

1. GEOMETRY BY: Harvandeep Shergill Ricmond Simran

2. Introduction • Hi , I am Harvandeep Shergill people call me Venus. I am in Grade 11 and 17 years old. Math is my favorite subject. • Hi I am Richmond Galima. I am in grade 11 and 17 years old. I would like to be a Nurse or a Health Care Aide. • Hi I am HarsimranKambo. I am a student of grade 11.I am 17 years old. I like to play badminton and work on computer.

3. Tasks • Tasks Completed by: • Harvandeep- • Equation of a circle •   Central Angle, Radii and Chord of a circle •   Properties of a polygon 2. Richmond • Circles in a coordinate Plane • 2. Properties,inscribed angles of a circle 3. Simran • Graphing of a circle • Tan-chord angle and Tan segments of a circle.  

4. CIRCLES CICLEs

5. HISTORY • The circle has been known since before the beginning of recorded history. It is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. • Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles. • Some highlights in the history of the circle are: • 1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 (3.16049...) as an approximate value of π. • 300 BC – Book 3 of Euclid's Elements deals with the properties of circles. • 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.

6. Equation of a circle • Remember that a circle is locus of points. A circle is all of the points that are a fixed distance, known as the distance, from a given point, known as the centre of a circle. • On a coordinate plane the formula becomes • (x-H)2 + (Y-K)2 =r2 •  h and k are the x and y coordinates of the centre of circle • H & K are also responsible for vertical and horizontal movement.

7. Circle In a Co-ordinate Plane • Circles, when graphed on the coordinate plane, have an equation of x2 + y2 = r2 where r is the radius (standard form) when the center of the circle is the origin.  When the center of the circle is (h, k) and the radius is of length r, the equation of a circle (standard form) is (x - h)2 + (y - k)2 = r2. Problem: Find the center and radius of (x - 2)2 + (y + 3)2 = 16. Then graph the circle. Solution: Rewrite the equation in standard form. (x - 2)2 + [y - (-3)]2 = 42 The center is (2, -3) and the radius is 4. The graph is easy to draw, especially if you use a compass. The figure below is the graph of the solution.

8. Graphing of Circle • Steps of graphing a circle with example. • x2 + y2 - 4x + 8y = 5 • First complete the squrares • (x-2)2 + (y-(-4))2 = 5+16+4 • (x-2)2 + (y-(-4))2 = (5)2 • Centre- (2,-4) • Radius- 5

9. Central Angle, Radii and Chord of a circle Radii of a circle Radii of a circle is any line originationg from the centre of a circle to the point on the circumference of a circle.Eg In the image shown below a , b , c are the radii. Central angle of a circle  Central angle is an angle formed by joining two points on the circumference of a circle to the center of the circle.Eg as shown below A,B are two points on circumference of the circle and O is the centre of a circle. Threfore Angle AOB is Central angle of this Cirlce Chord of a circle- Chord of a circle is a geometrical line segment whose endpoints joins curve of a circle.

10. Properties,inscribed angles of a circle • Inscribed Angles of A Circle • In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. • An inscribed angle is said to intersect an arc on the circle. The arc is the portion of the circle that is in the interior of the angle. The measure of the intercepted arc (equal to its central angle) is exactly twice the measure of the inscribed angle. • This single property has a number of consequences within the circle. For example, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal. It also allows one to prove that the opposite angles of a cyclic quadrilateral are supplementary.

11. Cont…………… •  Inscribed angles where one chord is a diameter • Let O be the center of a circle. Choose two points on the circle, and call them V and A. Draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B. • Angle BOA is a central angle; call it θ. Draw line OA. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ. • Angles BOA and AOV are supplementary. They add up to 180°, since line VB passing through O is a straight line. Therefore angle AOV measures 180° − θ. • It is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are: 180° − θψψ.Therefore • Subtract 180° from both sides, where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB. • Inscribed angles are which are frontal on the side of the circle

12. Tan-chord angle and Tan segments of a circle.   • Tangent Chord Angle: An angle formed by an intersecting tangent and chord has its vertex "on" the circle. • Tangent Chord Angle = 1/2 Intercepted Arc • Example: Find the measure of angle formed by the intersection of the tangent that intersects chord AC. • The measure of angle is half of the interceptedarc which is 170°. • Therefore x = ½ • 170 = 85° • What is the m of arc ABC? • m arc ABC= 2 • 70° =140°

13. Cont……. • Tangent Segment of a circle • Given a point outside a circle, two lines can be drawn through that point that are tangent to the circle. The tangent segments whose endpoints are the points of tangency and the fixed point outside the circle are equal. In other words, tangent segments drawn to the same circle from the same point (there are two for every circle) are equal. Example:  Find x in each of the following figures in Figure

14. Properties of a Polygon • Properties of polygon • Polygon-A closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex. • Properties • all sides equal • all angles equal. 

15. Proof Let there be a triangle ABC Draw a line from A to mid point of BC(X) i.e. perpendicular to BC

16. Two triangles are formed AXB & AXC In triangle AXB and AXC AX=AX (Common) Angle AXB = Angle AXC ( 90 degree each) CX=XB (X is Mid point) Therefore,  Triangle AXB ~= Triangle AXC So AC=AB Angle ACX = Angle ABX Draw a line from C to midpoint of AB(Y) i.e. perpendicular to AB

17. Two triangles are formed BCY & ACY In triangle BCY and ACY CY=CY (Common) Angle CYB = Angle CYA ( 90 degree each) BY=AY (Mid point) Therefore,  Triangle BCY ~= Triangle ACY So CB=AC Angle CBY = Angle CAY Draw a line from B to midpoint of AC(Z) i.e. perpendicular to AC

18. Therefore AC=AB , AC=AB , AB=BC So AB=BC=AC (all side to polygon ABC are equal) Also Angle ACB=Angle ABC, Angle CAB=Angle CBA, Angle BAC=Angle BCA So Angles ACB=ABC=BAC (Proved all angles of Polygon are equal) Hence Proved Two triangles are formed AZB & CZB In triangle AZB and CZB BZ=BZ (Common) Angle AZB = Angle CZB ( 90 degree each) CZ=ZA  (Mid point) Therefore,  Triangle AZB ~= Triangle CZB So BC=AB Angle ZCB = Angle ZAB

19. Source • www.wikipidia .com • www.google.com • www.google.com/images • www.maplesprecalculus.pbworks.com