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SUBMITTED BY: Name: MUHAMMAD RAFIQUE SKB No.: 136 Department: Mathematics Section : Secondary SHEIKH KHALIFA BIN ZAID COLLEGE PRACTICAL GEOMETRY Presentation.

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Presentation on theme: "SUBMITTED BY: Name: MUHAMMAD RAFIQUE SKB No.: 136 Department: Mathematics Section : Secondary SHEIKH KHALIFA BIN ZAID COLLEGE PRACTICAL GEOMETRY Presentation."— Presentation transcript:

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2 SUBMITTED BY: Name: MUHAMMAD RAFIQUE SKB No.: 136 Department: Mathematics Section : Secondary SHEIKH KHALIFA BIN ZAID COLLEGE PRACTICAL GEOMETRY Presentation

3 PRACTICAL GEOMETRY Introduction: Practical Geometry will enable the Student to identify and classify geometric figures and apply geometric properties to practical situations, special reasoning and visualization skill will be enhanced by using Geometric Models and a variety of technologies.

4 LESSON PLAN DATE: 27-07-11 Lecture No. 1 SUBJECT: Mathematics TOPIC:Practical Geometry CLASS/GROUP:X / IV NAME:Muhammad Rafique DA, SKBZ College LEARNING OBJECTIVES: How to Construct “Circum-Circle” of a triangle ABC? How to Construct “Escribed-Circle” opposite to the vertex “A” of the triangle ABC? How to Construct the “In Circle” (Inscribed Circle) of the  ABC? How to draw the “Medians” of a  ABC? How to draw the Altitudes of a triangle?

5 MOTIVATION : The teacher asks the following questions. How do you measure an angle? Do you make an angle of 60° with the help of compass or Protractor? How you construct a triangle when its three sides are given? How you construct a triangle when the measures of two sides and the included angle is given? How you construct a triangle when the measures of two angles and the included side is given?

6 REVISION OF THE PREVIOUS KNOLWEDGE What is a line and a line segment. What is a triangle? Sum of the measures of three angle of a triangle is …… Do the sum of the measures of two sides of a triangle is greater than the third side? How you bisect an angle? How you bisect a line segment? What are the vertices of a triangle? RESOURCE MATERIAL / TRAINING AIDS: Board, Lesson Plan, Planner, Chalk, Duster Compass, Protractor, Ruler

7 METHODOLOGY: By adopting the method of learning by doing and lecture methods of teaching. The teacher will explain how to: (i) Bisect a line Segment. (ii) Bisect an angle. (iii) Make a triangle when. (a) Three sides are given (b) Two sides and the included angle is given. (c) When two angles and the included side is given

8 SHORT TERM GOALS : Students will gain more marks if they learn it with perfection. Understanding of concept well. LONG TERM GOALS : It will be helpful to students in professional studies like mechanical engineering, instrumentation and civil engineering etc

9 REQUISITES/LINKAGE WITH PREVIOUS TOPIC/LECTURE: The Bisectors of Angles of a given triangle. Given: ABC is a triangle Required : To draw the bisectors of angles A, B and C Construction: 1.Draw the  ABC 2.With point B as a centre draw an arc of any radius, intersecting the sides BC and BA at point L and M ANGLES BISECTORS OF  ABC 3. Take point L as a centre and draw an arc of any radius. 4.Now take M as a centre and with the same radius draw the another arc, which cuts the previous arc at point P 5.Join point P to B and produce it. Thus BP is the bisector of { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_9.jpg", "name": "REQUISITES/LINKAGE WITH PREVIOUS TOPIC/LECTURE: The Bisectors of Angles of a given triangle.", "description": "Given: ABC is a triangle Required : To draw the bisectors of angles A, B and C Construction: 1.Draw the  ABC 2.With point B as a centre draw an arc of any radius, intersecting the sides BC and BA at point L and M ANGLES BISECTORS OF  ABC 3. Take point L as a centre and draw an arc of any radius. 4.Now take M as a centre and with the same radius draw the another arc, which cuts the previous arc at point P 5.Join point P to B and produce it. Thus BP is the bisector of

10 CONSTRUCTION : 1.Draw the  ABC. 2.To draw right bisector of the side AB, with B as a centre and radius more than half of AB, draw arcs on either sides of AB. 3.Now with A as a centre and with the same radius, draw arcs on either sides of AB cutting previous arcs at P and Q. 4.Join P and Q Thus PQ is the right bisector of AB, 5.Similarly, repeat steps (2) to (4) and draw ST and LM, the right bisectors of BC and AC respectively. Hence PQ, ST and LM arc the required right bisector of the sides AB, BC and AC respectively of  ABC. Note: Right bisectors of  ABC pass through O. So they arc concurrent. RIGHT BISECTORS OF  ABC

11 COVERAGE OF THE LESSON ACCORDING TO ITS SYNOPSIS: 1. Construction of Circum Circle Construct the  PQR in which mPQ = mQR = 8cm, m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_11.jpg", "name": "COVERAGE OF THE LESSON ACCORDING TO ITS SYNOPSIS: 1.", "description": "Construction of Circum Circle Construct the  PQR in which mPQ = mQR = 8cm, m

12 CONSTRUCTION: Draw a line segment QR = 8cm. At point Q draw an { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_12.jpg", "name": "CONSTRUCTION: Draw a line segment QR = 8cm.", "description": "At point Q draw an

13 2. CONSTRUCTION OF ESCRIBED CIRCLE : 2. Construction of Escribed Circle Question: Construct  ABC in which mAB = 7.5cm mBC = 7.5cm, m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_13.jpg", "name": "2. CONSTRUCTION OF ESCRIBED CIRCLE : 2.", "description": "Construction of Escribed Circle Question: Construct  ABC in which mAB = 7.5cm mBC = 7.5cm, m

14 CONSTRUCTION: Draw a line segment BC = 7.5cm. At point B, draw an < CBX=30° From BX, cut off AB = 7.5 cm. Join A to C. Thus  ABC is constructed. Produce (extend) BC beyond C to BY Produce BA to BX. Making two exterior angles { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_14.jpg", "name": "CONSTRUCTION: Draw a line segment BC = 7.5cm.", "description": "At point B, draw an < CBX=30° From BX, cut off AB = 7.5 cm. Join A to C. Thus  ABC is constructed. Produce (extend) BC beyond C to BY Produce BA to BX. Making two exterior angles

15 3.CONSTRUCTION OF IN CIRCLE Question: Construct the  ABC, in which m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_15.jpg", "name": "3.CONSTRUCTION OF IN CIRCLE Question: Construct the  ABC, in which m

16 CONSTRUCTION: Draw a line segment BC = 7.5cm At point “B” draw an B. Draw the bisector CC’ of >C (et them intersect each other at point O. From point “O” draw OD perpendicular on BC. With Centre “O” and radius equal to OD draw a circle. The circle touches the three sides of  ABC Hence it is our required in-circle is constructed.

17 4. CONSTRUCTION OF MEDIANS OF A TRIANGLE Question: Construct a  ABC, such that mAB = 6.8cm, mBC = 8cm and mAC = 10.0cm. Draw the medians of  ABC. Median of a Triangle: Line segments joining, the vertices of a triangle to the mid points of the opposite sides are called medians of the triangle. Given: A triangle ABC, in which mAB = 6.8cm, mBC = 8cm and mAC = 10.0cm. Required: Draw the medians of  ABC.

18 CONSTRUCTION: Draw a line segment BC = 8cm From point B draw an arc of radius 6.8cm. From point C draw an arc of radius 10cm which intersects the previous arc at point A. Join A to B and A to C. Thus  ABC is constructed. Draw the mid point D of Side BC Draw the mid point E of Side AC Draw the mid point F of Side AB Join A to D, B to E and C to F. Thus AD, BE and CF are our required medians of  ABC, which meet in a point O. It may be noted that medians of every triangle are concurrent (i.e. meet in one point) and their point of concurrency called “centraid”, divides each of them in the ratio of 2:1. By actual measurement it can be prove that mAO = mBO = mCO = 2 mOD mOE mOF 1

19 MEDIANS OF  ABC

20 5.CONSTRUCTION OF ALTITUDES OF A TRIANGLE Question: Take any triangle ABC and draw its altitudes. Given: A  ABC Required: To draw altitudes of the  ABC Altitude: A perpendicular from the vertex of a triangle to the opposite side is called an altitude of the triangle.

21 CONSTRUCTION: 1.Draw a  ABC 2. Take a point A as centre and draw an arc of suitable radius, which cuts BC at points D and E. 3. From D as centre, draw an arc of radius more than ½ mDE. 4. Again from point E draw another arc of same radius, cutting first arc at point F. 5. Join the points A and F, such that AF intersects BC at point P. Then AP is the altitude of the  ABC from the vertex A. 6. Similarly, repeat the steps (2) to (5) and draw BQ and CR, the altitudes of  ABC from the vertices B and C respectively. Hence AP, BQ and CR are the required altitudes of  ABC, which meet in a point O. The points P, Q and R are called the feet of these altitudes. Note: It may be remember that the altitudes of a triangle are always concurrent. The point of concurrency is called the “ORTHO CENTRE”

22 ALTITUDES OF  ABC

23 ASSESSMENT / EVALUATION: For assessment of the students, the teacher will give them following construction. 1) Draw the right bisectors of sides of a triangle ABC, in which mBC = 6.2cm, mAB = 4.5cm and mAC 5cm. 2) Construct  ABC in which mAB = 4.5cm, mBC = 5.5cm and m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_23.jpg", "name": "ASSESSMENT / EVALUATION: For assessment of the students, the teacher will give them following construction.", "description": "1) Draw the right bisectors of sides of a triangle ABC, in which mBC = 6.2cm, mAB = 4.5cm and mAC 5cm. 2) Construct  ABC in which mAB = 4.5cm, mBC = 5.5cm and m

24 SUMMARY OF THE LESSON: 1) A circle which passes through the vertices of a triangle is called circum-circle. 2) A circle which touches the three sides of a triangle is called In-Circle. 3) The line segments joining the vertices of a triangle with the mid-points of the opposite sides are called “Medians” of the triangle. 4) The point of intersection of three medians is called “Centroid” and medians are concurrent. 5) The line segments which are perpendiculars from the vertices to the opposite sides of a triangle are called its “Altitudes”. 6) The point of intersection of three altitudes is called “Ortho Centre” and they are concurrent. 7) Construction of unique triangle is possible when the sum of measures of two sides is greater than the third side.

25 HOME ASSIGNMENT: Ex No. 9.3 Book 1 P.No232Q.No. 1,2,3,5 Ex No. 7.1 Book 2 Q.No. 3 Construct a  ABC in which m AB = 5cm, m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_25.jpg", "name": "HOME ASSIGNMENT: Ex No. 9.3 Book 1 P.No232Q.No. 1,2,3,5 Ex No.", "description": "7.1 Book 2 Q.No. 3 Construct a  ABC in which m AB = 5cm, m

26 ACTIVITIES (M.C.Qs) 1)The sum of measures of all the angles of a triangle is __________ degree. (a) 180 (b) 360 (c) 90 (d) 270 2)Two lines are said to be perpendicular on each other if they form a/an _____ angle. (a) acute (b) Obtuse (c) Reflex (d) Right 3)The right bisectors of a triangle are ________ (a) Concurrent (b) Not concurrent (c) Concurrent and passes through the same point (d) None 4)The bisectors of angles of a triangle are and the point of concurrency is called __________ (a) Incentre (b) Ortho Centre (c) Circum Centre (d) Centroid. 5)A circle which passes through the vertices of a triangle is called __________. (a) In-circle (b) Circum-Cricle (c) Escribed Circle

27 Page -2- 6)A circle which touches the three sides of a triangle is called __________ (a) Circum circle (b) Escribed circle (c) In-Circle 7)The point of intersection of three medians is called _____. (a) Ortho Centre (b) Centroid (c) None (d) In-Centre 8)The point of intersection of three altitudes is called ________ (a) Centroid (b) Ortho Centre (c) In-Centre (d) Curcum Centre 9)The line segments joining the vertices of a triangle with the mid-points of the opposite sides are called __________ (a) Altitudes (b) Medians (c) Perpendiculars (d) None 10)The line segments which are perpendiculars from the vertices to the opposite sides of a triangle are called ________ (a) Medians (b) Altitudes (c) Bisectors (d) None

28 TOPIC TO BE TAUGHT NEXT Ex. 7.2 Book 2 Page No. 159 TOPICS: TO BE TAUGHT NEXT: 1)To draw two tangents to a circle from a point outside circle. 2)Direct common tangents to the two given circles. 3) To draw Transverse common tangents to the two given circles. MCQs Answer Key: 1-(a) 1802. (d) Right 3. (c) Concurrent and passes through the same point. 4-(a) In-Centre 5. (b) Circum Circle 6. (c) In-Circle 7. (b) Centroid 8. (b) Ortho centre 9. (b) Medians 10. (b) Altitudes

29 DA, SKBZ COLLEGE LESSONPLAN DATE: :28-07-2011 SUBJECT: Mathematics TOPIC:Practical Geometry CLASS/GROUP:X / IV LECTURE NUMBER:II NAME:Muhammad Rafique DA, SKBZ College SKB - 136 LEARNING OBJECTIVES: To draw a “TANGENT” to a circle from a given point out side the circle. To draw the “Direct Common Tangents” to the two given circles. To draw the “Transverse Common Tangents” to the two given circles.

30 RESOURCE MATERIAL / TRAINING AIDS : Board Lesson Plan Planner Chalk Duster Compass Protractor Ruler

31 REQUISITES/LINKAGE WITH PREVIOUS TOPIC/LECTURE What is circle? What is centre of the circle? How you define a radius? What is diameter? What is chord? How you define tangent? How you draw a circle. When its radius is given. What is circumference of a circle? How to draw the right bisector of side (line segment)?

32 BODY: (MATHODOLOGY) By adopting the method of learning by doing and lecture methods of teaching, The teacher will explain how to: (i) Draw a circle when its radius is given. (ii) Bisect a line Segment. (iii) Draw a tangent to a circle at a point on it. (iv) Define a circle and its centre (a) Radius of the circle (b) Diameter of the circle (c) Chord of the circle (d) Tangent of the circle (e) Circumference of the circle

33 COVERAG OF THE LESSON: CIRCLE: A set of points of a plane which are equidistant from a fixed (given) point is called a circle. The fixed point is called the centre of the circle. Centre of Circle RIADIUS OF THE CIRCLE: The distance of any point of the circle from its centre is called “radius” of the circle. It may be noted. C A 2cm Radius is a number not a line segment. The line segment joining the centre with any point of the circle is called the “Radial Segment”. In figure we see a circle whose centre is C and radius 2cm and CA is the radial segment.

34 CHORD OF THE CIRCLE: The line segment whose end points are any two points of the circle is called a Chord of the circle. In fig. CD is the chord. DIAMETER: A chord which passes through the centre of the circle is called a diameter. In fig PQ is a diameter. CIRCUMFERENCE OF A CIRCLE: The length of the curve joining all points of a circle is called circumference of the circle. Mathematically it is denoted by C = 2   Where  is the radius of the circle and  = 22 or 3.14 7  : It is the ratio of circumference to its diameter is called . CD P Q

35 TANGENT: If there is only one point Common between a line and a circle, the line is called a tangent to the circle. OR A line which is perpendicular to the outer end of the radius is called a tangent. OR A line which touches the circle at only one point is called a tangent. In Fig. PQ is a tangent. SECANT: If there are two distinct points common to a line and a circle, the line is called a secant of the circle. In fig. DE is a secant. SEMI-CIRCLE: The portion of a circle intercepted by a diameter is called semi-circle. In the figure diameter AB divides the circle in two equal (congruent) parts. Each is a semi circle. D E P Q A B

36 TO DRAW TANGENTS TO A CIRCLE FROM A POINT OUTSIDE THE CIRCLE : 1- Question: Draw a circle of radius 2.5cm Take a point T at a distance of 6cm from the centre. Draw the tangents to the circle passing through T. Measure the lengths of the tangents and verify the answer by mathematical calculation. Given: A circle of radius 2.5cm. T is a point out side the circle at a distance of 6cm from the centre of the circle. Required: To draw tangents to the given circle from the point-T Construction: 1. Draw a circle of radius 2.5cm with centre O. 2. Take a point “T” outside the circle such that OT = 6cm. 3. Draw the mid point M of OT. 4. With centre “M” and radius OM or MT draw a circle S2 intersecting the given circle “S1” at points P and Q.

37 5. Join T with P and Q. Thus PT and QT are the required tangents. 6. On measuring of PT or QT we get 5.4 cm VERIFICATION: In right angled triangle  OPT. m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_37.jpg", "name": "5. Join T with P and Q. Thus PT and QT are the required tangents.", "description": "6. On measuring of PT or QT we get 5.4 cm VERIFICATION: In right angled triangle  OPT. m

38 TANGENTS TO A CIRCLE FROM A POINT OUT SIDE THE CIRCLE

39 TO DRAW THE DIRECT COMMON TANGENTS TO THE TWO GIVEN CIRCLE: 2- Question: Draw a circle of radius 3cm and 1.5cm. The distance between their centres is 6cm. Draw their direct common tangents. Measure each tangent and verify by mathematical calculation. Direct Common Tangents to Two Circles: Definition: If the points of contact of the common tangent with two circles are on the same side of the line segment joining their centres. Such common tangent is called Direct Common Tangent. Construction: 1. Draw the line segment AB = 6cm, where A and B are centres of the two circles. 2. With centre A and radius r 1 =3cm draw the first circle S 1 3. With centre B and radius  2 = 1.5cm draw the second circle S 2. 4. With centre “A” draw the third circle S 3 of radius  3 = 3-1.5 = 1.5cm) 5. Draw the mid-point “M” of AB. 6. With centre “M” and radius equal to AM or MB draw the fourth circle S 4 cutting the third circle S 3 at points C and C’. 7. Join A with C and produce it to meet the first circle S1 at point “D”. 8. Similarly, join A with C’ and produce it to meet the first circle “S 1 ” at point D’.

40 9. Join B to C and C’. 10. With centre D and radius equal to BC draw an arc on S 2 i.e. E 11. With centre D’ and radius equal to BC draw an arc on S 2 i.e. E’. 12. Draw DE and D’E’. 13. Also Join B to E and E’. Thus DE and D’E’ are our required direct common tangents”. On measuring we get DE = D’E’ = 5.8cm VERIFICATION: In  ACB m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_40.jpg", "name": "9. Join B to C and C’. 10. With centre D and radius equal to BC draw an arc on S 2 i.e.", "description": "E 11. With centre D’ and radius equal to BC draw an arc on S 2 i.e. E’. 12. Draw DE and D’E’. 13. Also Join B to E and E’. Thus DE and D’E’ are our required direct common tangents . On measuring we get DE = D’E’ = 5.8cm VERIFICATION: In  ACB m

41 DIRECT COMMON TANGENTS

42 TRANSVERSE COMMON TANGENTS: Definition: It the points of contact of the common tangents of two circles lie on the opposite side of the line segment joining the centres of the circles then their tangents are called Transverse Common Tangents. Question : Draw the transverse common tangents of the circles with radii 3cm and 2cm, when the distance between their centres is 6cm. Measure each tangent and verify by mathematical calculations. Given: r 1 = radius = 3cm r 2 = radius = 2cm. Distance between their centres = 6cm. Required: Draw Transverse Common Tangents to these circles.

43 Construction: 1. Draw a line segment AB = 6cm the distance between their centres. 2. With centre A, draw the first circle S 1 of radius r 1 = 3cm. 3. With centre B, draw the second circle S 2 of radius r 2 = 2cm 4. With centre A, draw the third circle S 3 of radius r 3 = r 1 +r 2 = 3 + 2 = 5cm. 5. Draw the mid point “M” of AB. 6. With Centre M and radius equal to AM or BM, draw the fourth Circle S 4. Which cutting (intersecting) the third circle S 3 at points C and C’ respectively. 7. Join A to C and A to C’ which intersect the first circle at D and D’. 8. Join B to C and B to C’ 9. With centre D and radius equal to BC draw an arc on the second circle S 2 i.e. E 10. With Centre D’ and radius equal to BC draw an arc E’ on the Second Circle S 2. 11. Draw DE and D’E’. 12. Also join B to E and B to E’

44 TRANSVERSE COMMON TANGENTS

45 Thus DE and D’E’ are our required Transverse Common Tangents. On measuring we get m DE = mD’E’ = 3.8cm VERIFICATION: In  ACB, m { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3431607/slides/slide_45.jpg", "name": "Thus DE and D’E’ are our required Transverse Common Tangents.", "description": "On measuring we get m DE = mD’E’ = 3.8cm VERIFICATION: In  ACB, m

46 TESTING / ASSESSMENT / EVALUATION: For assessment of the students, the teacher will give them following construction. 1) Draw a circle of radius 2cm. Take a point P at a distance of 5.5cm from the centre of the circle. Draw the tangents to the circle passing through point P. Measure the length of the tangents and verify by mathematical calculation. 2) Draw two circles of radii 2.5cm and 1.2cm, with the distance between their centres equal to 5cm. Draw the Direct Common Tangents to these circles. Also measure and verify by M.C. 3) Draw the Transverse Common tangents of circles with radii 2.5cm and 1.5cm, when the distance between their centres is 5.5cm. Measure each tangent and verify by M.C.

47 SUMMARY OF THE LESSON: 1) Set of points of a plane which are equidistant from a fixed point is called a Circle. 2) A chord which passess through the Centre of the Circle is Called a diameter. 3) A line which is perpendicular to the outer end of the radius is called a tangent. 4) Angle inscribed in a Semi-circle is 90° 5) The side opposite to 90° is called hypotenuse. 6) In right angled triangle, according to Pythagoras theorem: (Hypt.) 2 = (Perp.) 2 + Base) 2 BRIDGE: The above construction plays a vital role in study of mathematics of H.S.C. i.e. XI & XII. For the study of analytical Geometry. In deed, it is the back bone in study of Architectural and Civil engineering.

48 ACTIVITIES (M.C.Qs) 1)The line segment whose end points are any two points of the circle is called a __________ of the circle. (a) Chord (b) Diameter (c) Radius (d) Tangent 2)A chord which passes through the centre of the circle is called (a) Diameter (b) Radius (c) Tangent (d) Secant 3)If a line touches the circle at only one point is called a ________ (a) Tangent (b) Secant (c) Chord (d) Diameter 4)Two circles are said to be congruent if their __________ are equal (a) Radii (b) Chords (c) Diameters (d) Tangent 5)An angle inscribed in a Semi-Circle is _______ degree. (a) 90° (b) 180° (c) 60° (d) 360° 6)In right angled triangle, the side opposite to 90 degree is called _______. (a) Hypotenuse (b) Perpendicular (c) Base (d) Radius

49 Page -2- 7)A line which is perpendicular to the outer end of the radius is called a __________ (a) Tangent (b) Secant (c) diameter (d) Chord 8)Two ________ can be drawn to a circle from a point outside the circle. (a) Tangents (b) Secants (c) Diameters (d) Chords. 9)If the legs of a right angled triangle are 1, 1 then its hypotenuse is _________ (a)  2 (b) 2 (c) ½ (d) 1 10)The line which is at a distance equal to radius of a circle from its centre is ________ to the circle (a) Tangent (b) Radius (c) Diameter (d) Secant MCQs Answer Key: 1.(a) Chord 2. (a) Diameter 3. (a) Tangent 4.(a) Radii 5. (b) 90° 6. (a) Hypotenuse 7. (a) Tangent 8. (a) Tangents 9. (a)  2 10. (a) Tangent


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