MATH 30 GEOMETRY UNIT OUTLINE AND DEFINITIONS Prepared by: Mr. F.
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1 1 MTH 30 GEMETRY UNIT UTLINE ND DEFINITINS Prepared by: Mr. F. Some f The Typical Geometric Properties We Will Investigate: The converse holds in many cases too! The Measure f The entral ngle Tangent To ircle Is Perpendicular To Is Equal To Twice The Measure f The Radius t The Point f Tangency The Inscribed ngle Subtended y The Same rc r hord Inscribed ngles Subtended y The Same rc re ongruent The Tangent Segments To ircle, From ny External Point, re ongruent The ngle Inscribed In Semicircle (or on a diameter chord) Is Right ngle hords that are equi-distant from the centre of a circle are congruent hords that are congruent are equi-distance from the centre of the circle (converse) pposite ngles f yclic Quadrilateral re Supplementary The Perpendicular isector f hord ontains The entre f The ircle The Sum f The Interior ngles f n N-Sided Polygon Is (N-2) * 180. The Perpendicular From The entre f ircle To hord isects The hord (converse) M30S_H_GeometrylassNotes.doc Revd: 12 Nov 2008
2 2 Selected Vocabulary of Geometric Terms rc n arc of a circle is two points on the circle and the part of the circle between the two points. The two points are called the end points of the arc (Diagram 4). The Measure of an arc is the measure of its central angle. entral ngle hord ircle n angle that has its vertex at the centre of a circle and its end points on the circumference of the circle (Diagram 5). chord of a circle is a segment whose endpoints are on the circle (Diagram 2). circle is the set of all points in a plane equidistant from a given fixed point. ircumference The circumference is the distance around a circle (Diagram 1). omplementary ngles ngles whose sum is 90 oncentric ircles ongruent ongruent ircles onverse orollary yclic Quadrilateral Diameter If two or more circles share the same centre, then they are concentric circles (Diagram 3). In geometry, congruent means the same as equals, except it is broader in meaning; it includes shape, size, not just length. If two circles have the same radius, they are congruent circles (Diagram 3). onverse of a statement is changing the position of the assumption and the conclusion. Example: If you have brown eyes then you are cute. The converse is : If you are cute then you have brown eyes. The converse is not always true in real life, but it often is in geometry! - a theorem that is easily proved from a first theorem. Like a subtheorem or a special case quadrilateral whose vertices are concyclic (points on the same circle) is called a cyclic quadrilateral. This feature may be described as a quadrilateral inscribed in a circle or as a circle circumscribed about a quadrilateral (Diagram 7). diameter of a circle is a chord that passes through the centre (Diagram 2).
3 3 Equidistant Equilateral triangle Inscribed ngle Intercepted rc Isosceles triangle the same distance from something - a triangle whose sides are all equal in length. onsequently, the angles are all equal (60 ) too! n angle that has its vertex on the circle. n angle formed by two chords that intersect on the circle, each with an endpoint at the vertex of the angle (Diagram 5). n arc that lies in the interior of an angle with one endpoint on each side of the angle (Diagram 5). - a triangle with two sides of equal length. onsequently, the angles opposite the congruent sides are congruent also Length = Length, so the triangle is isosceles. Further, = Major rc Minor rc Radius (Plural: Radii) Secant Sector Segment major arc is an arc of a circle that is larger than a semicircle (Diagram 4). minor arc is an arc of a circle that is smaller than a semicircle (Diagram 4). The radius of a circle is a segment that has the centre as one end point and a point on the circle as the other endpoint. The radius can be referred to as a line segment or as a length of a line segment (Diagram 1). secant is a line that intersects a circle at two points (Diagram 2). The sector of a circle is a region bounded by two radii of a circle and the intercepted arc. Sectors can be minor, major, or half circles as determined by the minor, major, or semicircle arcs that they intercept (Diagram 6). segment of a circle is the region bounded by a chord and its intercepted arc. Segments can be classified as minor, major, or half circles as determined by minor, major, or semicircle arcs (Diagram 6).
4 4 Semicircle Subtended ngle semicircle is an arc of a circle whose end points are the endpoints of the diameter (Diagram 4). The angle formed at a point by line segments from that point connected to a chord or an arc. (Diagram 8) Supplementary ngles 2 angles whose measures, when added together, equal 180 degrees. Z Z is supplementary to Z. Tangent Vertex of an angle tangent is a line that intersects the circle at only one point. The point where the tangent touches the circle is called the point of tangency (Diagram 2). - the common endpoint of two rays. The pointy part of an angle. DIGRMS DIGRM 1 DIGRM 2 DIGRM 3
5 5 DIGRM 4 DIGRM5 The angles intercept an arc The arc subtends the angles at a point DIGRM 6
6 6 DIGRM 7 DIGRM 8 The angle is said to be the angle subtended by the minor arc (or simply arc ) at the centre above. The angle is said to be the angle subtended by the minor arc (or simply arc ) at
7 7 1. Here are several ideas that you will need to recall from Middle School and Grade 10 Geometry to help with your solving of Grade 11 Geometry. 2. Number of degrees in a triangle. Every triangle has three corners. The sum of the measures of the angles of each corner is 180 The sum of all three corners of a triangle always equals Number of degrees in a 4 sided polygon. The number of degrees in a 4 sided polygon is Properties of Isosceles triangles. Remember that an isosceles triangle has two sides the same length, which also means it has two angles the same! The Sine Law proves that. 5. straight line or line segment can be thought of as a 180 angle around a point on that line. In an isosceles triangle two sides are the same length and the two angles opposite them are the same measure. In the diagram at left side = side and = If equilateral triangle then all three sides and angles are the same 180
8 8 6. Pythagoras Theorem The square of the longest side of a right angle triangle equals the sum of the squares of the other two sides. c a c 2 = a 2 + b 2 7. Transversals and pposite ngles b Given a single line or two parallel lines cut by another line the angles marked 1 are all equal, the angles marked 2 are all equal. lso notice that opposite angles of intersecting lines are congruent. ENTRL ND INSRIED NGLES IN IRLES 1. Recall Definitions: entral ngle: Inscribed ngle: ngle with vertex at center, endpoints on circumference. ngle with vertex on circle, endpoints on circle. Intercepted rc n arc that lies in the interior of an angle with one endpoint on each side of the angle. We say the angle is subtended by the arc or the arc is intercepted by the angle. yclic Quadrilateral quadrilateral whose vertices are concyclic (points on the same circle) is called a cyclic quadrilateral. This feature may be described as a quadrilateral inscribed in a circle or as a circle circumscribed about a quadrilateral 2. Property: If a central angle and an inscribed angle intercept the same chord or arc, the central angle is twice the inscribed angle. Z m PQ = 2 * m PZQ P Q
9 9 3. orollary: The angle inscribed in a semicircle (ie: along a diameter) is a right angle. The converse is true also: if a right angle is inscribed in a circle then the angle must intercept a diameter of the circle. Z 4. corollary is just a sub-rule special case that is evident from a more general rule. m PQ = 180 So m PZQ = ½* 180 = 90 P Q Many lacrosse and hockey goalies understand this rule! 5. Every carpenter understands this corollary also. It is how they find the diameter and centre of a circle using a arpenters Square! 6. Property: Inscribed angles subtended by the same arc are equal. m TPZ = m TQZ since they are inscribed angles that share the same arc TZ. P Q T 7. What can you say about angles PZQ and PTQ? Will the converse statement work??? Z
10 10 Examples 8. Determine the measure of angle x: x is at the centre of the circle, angle D = Find the measure of angles D and D. m D = m D = D pposite angles of a cyclic quadrilateral are supplementary 11. Given measure of angle = 80, what is the measure of angle D? m D = D 12. What does supplementary mean?
11 Given: is a diameter. D is 50. D is 80. Find and explain why: a. m D D b. m
12 12 PRPERTIES F PLYGNS 1. PLYGN: (Definition) closed figure formed by three or more line segments 2. omplete the following table knowing that there is 180 in the interior angles of a triangle. Nbr of Sides of Polygon (n) n 2 Small Diagram Number of Triangles Sum of interior ngles of the Polygon Properties of Polygons. For any polygon with n sides, the sum of the angles is (n-2)* Practice. omplete the following table: Polygon Number of Triangles Sum of Interior ngles Rhombus 9 sided polygon 30 sided polygon ctagon Pentagon
13 13 PRPERTIES F TNGENTS ** Recall : line that is tangent to a circle, touches the circle at only one point, the point of tangency. 1. Property. tangent to a circle is perpendicular to the radius at the point of tangency. (The converse applies also). P Q Example 1: 2. Given a circle with centre having a radius of 5 cm, line PQ is tangent to the circle, and length PQ is 12, find length Q. P Q 3. Find the area of triangle PQ. (rea = ½ * base * height) Example 2: 4. is tangent to circle with centre. =, = 6, angle = is to Find the area of figure. What is the length of and why? What type of triangle is?? (Recall the area of a trapezoid? If not just make three triangles)
14 14 5. Example 3. Given: = 5 and are tangents of the circle at and respectively. ngle D = 30 0 ngle = 50 0 Find the length of D and why?: D Needs Sine Law! Pretty advanced Question! 6. Property: Tangent segments to a circle, from any external point, are congruent. P Z Q 7. Example. In the above diagram Lines PZ and QZ are tangent to the circle of radius 5cm. Lines PZ and QZ meet at an external point Z. The distance from P to Z is 10cm. a. find length QZ: c. find the area of quadrilateral PZQ: d. what is m PZQ? b. find length Z: e. if this was a side view of a cone sitting on a sphere, what would the volume of the cone be? (advanced question!)
15 15 hord Perpendicular isector Theorems. Properties: (summarized) 1. line from the centre of a circle, perpendicular to a chord, bisects the chord. PRPERTIES F HRDS F IRLE a. The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord. b. The perpendicular bisector of a chord passes through the centre of the circle. 4. ongruent chords equidistant from centre. a. ongruent chords are equidistant from the centre of the circle. a. hords equidistant from the centre of a circle are equal. line from the centre of a circle, perpendicular to a chord, bisects the chord 6. Line Z is drawn perpendicular to the chord PQ. It cuts PQ in exactly half (bisects the chord). 7. Given: is the centre of the circle, radius is 10 cm, length Z is 6 cm, find length PZ. Why is it that? P Z Q
16 16 The line segment drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. 8. If we draw any chord and cut it in half (bisect it) then when we connect that mid-point to the centre, we form a 90 angle with the chord. 9. onstruct it! Make any chord. Find the midpoint of the chord. onnect that to the centre with a line. Measure the angle of the chord with the line through the centre and the midpoint of the chord; it is This is really a corollary of the previous property. The perpendicular bisector of a chord passes through the centre of the circle. 11. onstruct it yourself. Make any chord anywhere. ut it in half, find the midpoint. Draw a line at a 90 angle through the mid-point. The line will go exactly through the centre of the circle. 12. This is another way that carpenters find the centre of a circle! 13. Do you know how to bisect a line segment without a ruler? Do you know how to make a 90 angle without a ruler? (Napoleon did; he was an amateur geometrist!)
17 17 ongruent chords are equidistant from the centre of the circle. 14. Have you every played in a large drainage pipe? Every tried to squeeze a 2 by 4 trough it or a sheet of wood? 15. You construct it! Draw any two chords of the same length on the circle. Measure their distance to the centre! (f course when you measure distance to something you always measure it the shortest perpendicular distance). Label all the points! orollary and converse: hords equidistant from the centre of a circle are equal. 16. Same as the above property but the converse. Examples. 17. is the centre of the circle. Find length D. Why? If the radius is 10 cm what is the distance from the centre to the chord? Why? D
18 18 Practice Problems for the Entire Unit 1. Given; is centre of the circle; = 5 and = 110 Find and state why: a. m b. m D c. m d. m D e. Length
19 19 2. Given: entre at, Diameter = 10 and D = 50 Find and state why: a. m D b. m D c. Length D d. Length D e. m D
20 20 3. Given: entre, Diameter =10, Z=120, and Length = 6. Find and state why: a. m b. Radius c. Length Z d. rea of the Semi-ircle T e. rea of f. rea of Sector g. Length of rc
21 21 4. What is the Total Sum of the internal angles of these polygons? 5. Given: entre, D = 90, D=50, DGE = 70 Find and state why: a. m D D b. m D G c. m ED E d. m G
22 22 6. Given: entre, X= Y =3, = 8 Find: a. Length DY X b. Length D c. Radius of the circle d. Length E E Y D e. Measure of ngle E
23 23 7. Given: = 10, entre, = 110 X Find and explain why : a. Length X b. D 12 5 Y D c. Radius
24 24 8. Given: Tangents and, =12, =13. Find and explain why: a. Radius b. Find If you want to know more about this geometry and the actual proofs of the rules, do some research on a famous Greek fellow named Euclid
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