chapter 3

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chapter 3

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Keywords:politeknik kuching sarawak, reflex angle, supplementary angle, complementary angle, straight angle, right angles, obtuse angle, opposite angles, mathematics science, acute angle, aoy, rqs, computer department, pqs, aob, pqr, right angle, endpoint, straight line, chapter 3

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Date Added:Oct 26 2009 7:51PM

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#font 2972 "AODCGA+TimesNewRomanPS ItalicMT 287 0" #font 1990 "AOMKMN+ArialMT 280 0" #font 1920 "AOCOCP+TimesNewRomanPSMT 283 0" #font 1579 "AOKHMA+Arial ItalicMT 277 0" #font 1195 "AOHOGP+Arial ItalicMT 273 0" x x A B C D A B E D C A T B A O C x #font 45 "AODDJI+MT Extra 369 0" o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o #font 41 "AODDIN+TimesNewRomanPS BoldItalicMT 364 0" A revolution A straight angle A right angle An acute an........

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#font 2972 "AODCGA+TimesNewRomanPS ItalicMT 287 0" #font 1990 "AOMKMN+ArialMT 280 0" #font 1920 "AOCOCP+TimesNewRomanPSMT 283 0" #font 1579 "AOKHMA+Arial ItalicMT 277 0" #font 1195 "AOHOGP+Arial ItalicMT 273 0" x x A B C D A B E D C A T B A O C x #font 45 "AODDJI+MT Extra 369 0" o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o #font 41 "AODDIN+TimesNewRomanPS BoldItalicMT 364 0" A revolution A straight angle A right angle An acute angle An obtuse angle A reflex angle Complementary angle Supplementary angle Vertically Opposite Angles Solution: Solution: Solution: Solution: Solution: Solution a : Solution b : Solution: Solution: In any right angled triangle , the square on the hypotenuse is equal to the sum of the squares on the other two sides. Or Solution: Solution: Solution: Solution: Solution: Solution: Solution: Solution FOR : Solution FOR : Solution: Solution: Solution : Solution a : Solution b : Solution c : Solution d : Perimeter and Area Rectangle Parallelogram Trapezium Circle Triangle Solution a : So, Perimeter of the semicircle Solution b : So, Area of the semicircle , Solution a : So, Perimeter of the rectangle Solution b : So, Area of the rectangle, Area and Volume Cube Cuboid Cylinder Pyramid Cone Sphere Solution a : So, Area of the cuboid, Solution b : So, Volume of the cuboid , Solution a : So, Area of the cylinder, Solution b : So, Volume of the cylinder, similar shapes legal #font 39 "AODDEF+ArialMT 363 0" A G F B D H E C Centre Radius Diameter Chord Minor arc Minor segment Major arc Major segment Sector O N M D C B A O M O Q P 4 cm M O Q P 8 cm 3 cm N O Q P C C O A O C B O S R Q P T S R Q P C O O C B A T S R Q P iii ii i O B A T O B A T O B 50 O B A T 46 T A O B T A O B rad @ O x 78 y 67 O 40 O 180 O 92 C B A O O T A B 70 C T A B 46 C O T B A 35 O T A B 30 O B A T 240 , = , = , = = = Cone 1 Cone 2 Cone 1 Cone 2 150.2 mm 75.1 mm #font 30 "AODDAH+SymbolMT 362 0" + = = = + = + = + + + = + = = = = + + = + = = = = = = = = = = = = = = = = + + = + = = = = = = + = = + = = + = = + = + + = = + = = = = + + = = = = = = = = + + = = + = = = = = + + = = = = + = = = + = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = + + + = = + + + = = + + + = + = = = + + = = = = + = = = = + = = = = = = + + = = + = = + = = + = = = + + = = + + = = = = + = + = = = = = = = = = = = = = = = = = = = = #font 22 "AODCOI+TimesNewRomanPS BoldMT 350 0" CHAPTER 3: GEOMETRY AND MEASUREMENT B1001 3.1 ANGLE Definition of Angle 3.1.1 Types of angle E XAMPLE 3.1 3.1.2 Angle related to intersection and parallel lines E XAMPLE 3.2 TUTORIAL SUBTOPIC 3.1 3.2 CIRCLE Definition of circle: 3.2.1 Essential parts in a Circle 3.2.2 Theorem of Pythagoras hypotenuse theorem of Pythagoras E XAMPLE 3.3 3.2.3 Chord in a circle E XAMPLE 3.4 3.2.4 Angle in a circle E XAMPLE 3.5 3.2.5 Tangent to a circle E XAMPLE 3.6 3.26 Length of arc, area of a sector and segment in degree or radian degrees radians E XAMPLE 3.7 E XAMPLE 3.8 TUTORIAL SUBTOPIC 3.2 O O 3.3 PERIMETER, AREA AND VOLUME 3.3.1 of various 2D shapes 2D Shape Perimeter Area E XAMPLE 3.9 3.3.2 of various 3D shapes 3D Shape Area Volume E XAMPLE 3.10 3.3.3 Relationship between corresponding sides, area and volume for . TUTORIAL SUBTOPIC 3.3 #font 18 "AODCLD+Arial BoldMT 349 0" POLITEKNIK KUCHING SARAWAK MATHEMATICS, SCIENCE AND COMPUTER DEPARTMENT #font 16 "AODCGB+TimesNewRomanPS ItalicMT 348 0" 1 AOB Figure 3.1 A revolution angle = A straight angle = AOB OX AOX XOB AOX BOX A right angle = An acute angle An obtuse angle A reflex angle : : BOY AOX AOY BOX BOY AOX AOY BOX PQR Since, PQT and RQT is equal to PQS is greater than and less than RQS is greater than and less than Hence, PQT and RQT are Right Angles PQS is an Acute Angle RQS is an Obtuse Angle As PQR is straight line, so addition of angles PQS RQS Hence, Since, Addition of A BD and CBD So, A BD and CBD is supplement to each other. Since, Addition of PQT and SQT So, PQT and SQT is to each other. Since, Here we have an angle of revolution So, An obtuse angle Since, Here we have a straight angle AOB So, An acute angle Since, Here we have vertically opposite angles where, So, An acute angle Mathematically parallel to each other marked with arrow sign Figure 3.2 cut by a line transversal Transversal G E C A AND H F D B Since, Here we have parallel lines line AB parallel with line CD Then, MFG NFG So, Since, Here we have parallel lines line AB parallel with line CD and a transversal TW Then, BUT and DVT BUT So, Radius Chord iii. Diameter: Arc Segment Sector Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Theorem of Pythagoras a must Figure 3.3 c b a BC. By Pythagorass Theorem, c b a So, c c c c c cm BC. By Pythagorass Theorem, e f d So, d d d d d cm M B A Properties on Chords Properties 1 : AB, MB Properties 2 : AB CD OM ON x Since, OM perpendicular to PQ Then, PQ MQ So, x cm Since, NQ perpendicular to ON So, = 90 x Since, Here we can find value of x using theorem of Pythagoras. and MQ = 4 cm So, OQ MQ OM x x x x cm B A Properties on Angle Properties 1 : AOB ACB Properties 2 : AB ACB Properties 3 : ADB ACB ABDC ACB ADB Properties 4 : P R S Q Properties 5 : RST PQR x Since, According to properties of angle in circle, AOB ACB So, x x x Since, ACB So, x x Since, RST PQR So, Since, So, x x Figure 3.8 When a straight line touches a circle, it is called a tangent to the circle at that point Properties 1 : TAB OA TA Properties 2 : TA TB T TA TB TOA TOB ATO BTO Properties 3 : TAB A TAD ACD BAC ADC a b a b a Since, TB perpendicular to OT Then, BTO So, a a Since, TA = T B So, a cm , b = 30 Since, TB is a tangent to circle So, a b rad rad radian to degree rad degree to radian rad Figure 3.9 d. Length of arc in a circle r s e. Area of sector r A tor f. Area of triangle r A Triangle g. Area of segment shaded region segment A Triangle tor A A segment A r r segment A r Since, r s So, Since, r A tor So, tor A tor A cm Since, r A Triangle So, Triangle A Triangle A cm Since, segment A r So, segment A segment A cm F x x x using theorem of Pythagoras N A L I J x x and y x x x x y x x y y y x inches OAB w w l l w l a a b b h b c a b b h b b r r r c b a h b l w h b a b b h c a a c Since, r s or rad cm r Then, s = = cm cm Since, r A semicircle A cm A Since, Perimeter = 2l + 2w cm Since, w l A gle rec gle rec A cm x x zy xz xy x y z rh r h r bc ab abh rl r h r r r surface area volume Since, cuboid Azy xz xy cuboid A cm Since, cuboid V x yz cuboid V cm volume surface area Since, cylinder A rh r cylinder A cm Since, cylinder Vh r cylinder V cm congruence to figures Figure 3.10 r r h h r r As As h h r r As As r r V V r r V V h h h h r r A A V V a, 2a a a and . of the shape is 12cm 2 . volume cm h and cm r cm r h #font 3 "AOCOPI+TimesNewRomanPSMT 347 0" Page 1 of 29 gagaulih@jeffery_jmsk2008 An angle consists of two lines with a common endpoint. The two lines OA and OB in the figure 3.1 form an angle named . a. is the angle formed by rotating a line about its endpoint once until it comes back onto itself. A revolution is defined as to measure 360 . 360 b. is the angle formed by a line. A straight angle is half a revolution and so measures 180 . 180 c. is half of a straight angle and so measures 90 . Suppose that is a lines and that is a line such that is equal to . Then and is called a right angle. 90 d. is an angle greater than 0 and less than 90 . 0 90 Page 2 of 29 gagaulih@jeffery_jmsk2008 e. is an angle greater than 90 and less than 180 . 90 180 f. is an angle greater than 180 and less than 360 . 180 360 g. Two angles are called complementary if they add to 90 . For example 15 is complement of 75 . h. Two angles are called supplementary if they add to 180 . For example 105 is supplement of 75 . i. : Each pair of opposite angles formed when two lines intersect is called vertically opposite angles. In figure below, and are vertically opposite. The unmarked angle and are also vertically opposite. = = 1. Given that is a straight line in the figure below. a List down an acute angle, an obtuse angle and right angles. 90 90 90 90 180 Page 3 of 29 gagaulih@jeffery_jmsk2008 b Find value angle of and state type of the angle. and is 180 . 75 180 180 75 105 2. List down two angles that supplement to each other. is 180 130 50 180 3. List down two angles that complement to each other. is 90 40 50 90 complement 4. Find in the figure below and state type of the angle. 360 30 90 110 360 230 360 230 360 130 50 Page 4 of 29 gagaulih@jeffery_jmsk2008 5. Find and in each figure below state type of each angle. a b 180 3 90 2 180 5 90 180 5 90 5 90 18 3 120 3 120 3 120 40 a. Figure 3.2 shows two lines marked with AB and CD. b. , both lines are if they are as indicated in figure 3.2. c. If two parallel lines are , then Figure 3.3 Page 5 of 29 gagaulih@jeffery_jmsk2008 1. Find in the figure below. 110 120 120 110 360 230 360 130 2. Find value of and 130 130 130 130 180 50 Page 6 of 29 gagaulih@jeffery_jmsk2008 1. Find the angles or in figure below and state type of each angle. a b c d e f g h 2. Find the angles or in figure below and state type of each angle. a b c d e f g 3. Find the angles , , or in figure below and state type of each angle. a b c d e f g h 4. Find the angles or in figure below and state type of each angle. Page 7 of 29 gagaulih@jeffery_jmsk2008 a b c d e f g h 5. Find the angles , or in figure below and state type of each angle. a b c d e f g 6 a Show that 180 b Show that c Show that Page 8 of 29 gagaulih@jeffery_jmsk2008 A circle is a closed curve in a plane such that all points on the curve are equidistant from a fixed point. Figure 3.4 i. : The given distance is called the radius of the circle. ii. : A chord is a line segment with its end points on the circle. A diameter is a chord passing through the centre. iv. : An arc is a part of the circle. v. : A segment is the region bounded by a chord and an arc of the circle. vi. : A sector is the region bounded by two radii and an arc. Page 9 of 29 gagaulih@jeffery_jmsk2008 a. : One of utmost important theorem related to geometry which is to learn before we have a further study in geometry. b. With reference to Figure 3.3, the side opposite the right angle i.e. side c is called the . The states: 2 2 2 1. In figure below, find the length of 2 2 2 2 2 4 3 2 2 16 9 2 25 25 5 2. In figure below, find the length of 2 2 2 2 2 5 2 13 2 2 2 5 13 2 144 144 12 Hypotenuse Page 10 of 29 gagaulih@jeffery_jmsk2008 a. of a circle The line joining the centre to the midpoint of a chord is perpendicular to the chord. If OM perpendicular then MA = Equal chords are equidistant from the centre of a circle. If = , then = Page 11 of 29 gagaulih@jeffery_jmsk2008 1. O is the centre of the circle. Find value of length in each of the following figures. 4 2. Find value of angle . 3. Find value of angle . 2 2 2 2 3 4 3 2 2 16 9 2 25 5 Page 12 of 29 gagaulih@jeffery_jmsk2008 a. of a circle The angle that an arc of a circle subtends at the centre is twice the angle that it subtends at any point on the remaining part of the circumference. If O is the centre of the circle, then = 2 The angle in a semi circle is a right angle. If is a diameter, then =90 Angles in the same segment of a circle are equal. If and are in the same segment , then = The opposite angles of a cyclic quadrilateral are supplementary. + = 180 + = 180 If one side of a cyclic quadrilateral is extended, the exterior angle equals the interior opposite angle. = Page 13 of 29 gagaulih@jeffery_jmsk2008 1. O is the centre of the circle. Find value of angle in following figure. = 2 2 120 60 2. O is the centre of the circle. Find value of angle in following figure. 90 50 90 180 40 3. O is the centre of the circle. Find value of angle and in following figure. 96 88 180 92 120 50 96 88 Page 14 of 29 gagaulih@jeffery_jmsk2008 a. Definition of a tangent to a circle: b. Figure 3.8 shows the three possibilities that a straight line i. does not intersect a circle ii. intersects a circle at two points iii. touches a circle intersects at one and only point . c. . The following theorem states a basic property of a tangent to a circle. The tangent to a circle at a point is perpendicular to the radius at that point If is a tangent at A, then perpendicular If two tangents are drawn to a circle from an external point, a the tangents are equal; b the tangents subtend equal angles at the centre; c the line joining the external point to the centre bisects the angle between the tangents. If , , are tangents from , then = = = The angles between a tangent and a chord through the point of contact are equal respectively to the angles in the alternate segment. If , is a tangent at , then = = Page 15 of 29 gagaulih@jeffery_jmsk2008 25 30 5 cm 1. TB is a tangent to the circle at points A. Find the unknowns in each of the following figures. a 90 25 90 65 b 5 c 50 46 Page 16 of 29 gagaulih@jeffery_jmsk2008 r Radius s Arc Angle in radian a. Measure angle in degree and radian i. At school we usually learn to measure an angle in . However, there are other ways of measuring an angle. ii. One way that we are going to have a look at here is measuring angles in units called . 180 1 and 180 1 , where 142 . 3 iii. To convert 180 iv. To convert 180 1. Fill in the blank with given angle in degree. Use 142 . 3 . Angle in radian rad 0.26 Angle in degree 15 30 45 60 75 90 180 270 360 2. Fill in the blank with given angle in radian. Use 142 . 3 . Angle in radian rad 0.3 0.5 0.6 0.8 1.0 3.0 5.0 6.0 Angle in degree 17.2 b. Calculation of length of arc, area of sector and area of segment requiring a strong understanding of certain formula. c. With reference to figure 3.9, 4 main formulas are utilized. Page 17 of 29 gagaulih@jeffery_jmsk2008 Length of arc of a circle is given by Area of a sector of a circle is given by 2 sec 2 1 Area of a triangle of a circle is given by sin 2 1 2 Area of a segment of a circle is given by sec sin 2 1 2 1 2 2 sin 2 1 2 1. Consider the circle shown in figure below. a. Calculate the angle . b. Area of sector AOB c. Area of triangle AOB d. Area of shaded region 25 10 10 25 2 5 22 . 144 2 sec 2 1 sec 2 5 10 2 1 2 sec 2 125 Page 18 of 29 gagaulih@jeffery_jmsk2008 sin 2 1 2 22 . 144 sin 10 2 1 2 2 94 . 29 sin 2 1 2 22 . 144 sin 2 5 10 2 1 2 2 77 . 95 Page 19 of 29 gagaulih@jeffery_jmsk2008 M S R Q P 2 cm cm N 1. Find the value of . a b c d 12 cm 9 cm 5 cm cm 2. Find the value of . a b c d e f Page 20 of 29 gagaulih@jeffery_jmsk2008 84 38 20 120 y g h i j k l m n o Page 21 of 29 gagaulih@jeffery_jmsk2008 3. In a circle, find area and length of arc, if the radius is 15.8. Use 3.14 for . 4. In a circle, find area and length of arc, if the diameter is 1 2. 3 use 22 7 for . 5. In a circle, find area, radius and diameter, if length of arc is 28 cm. 6. A semicircle has an arc length of 20 cm. Find the radius of the semicircle. Hint: Draw a sketch of the semicircle. 7. Find the a. Perimeter of the figure below. b. Area of the figure below 5cm. 8. A circle with a centre O is shown in figure below. Find the angle a. The area of the sector , b. The area of the minor segment shaded . 9. Calculate the length of wire in the paper clip shown in figure below. The dimensions are millimeters. Page 22 of 29 gagaulih@jeffery_jmsk2008 In this section we will discover the perimeter and area formulas of 2D shapes. Perimeter Area Perimeter Area Perimeter 2 1 Area 2 1 2 1 Perimeter Circumference 2 or Area 2 2 1 Perimeter Area 2 1 1 2 Page 23 of 29 gagaulih@jeffery_jmsk2008 1. Consider the semicircle with radius 12 cm shown in figure below. Calculate a. The perimeter b. The area 180 6 6 85 . 18 12 85 . 18 85 . 30 2 2 1 2 6 2 1 2 56 . 56 2. Calculate a perimeter and b area for figure below. 4 2 6 2 20 tan tan 2 24 4 6 Page 24 of 29 gagaulih@jeffery_jmsk2008 Area 2 6 Volume 3 Area 2 2 2 Volume Area 2 2 2 Volume 2 Area 2 Volume 3 1 Area 2 Volume 2 3 1 Area 2 4 Volume 3 3 4 Page 25 of 29 gagaulih@jeffery_jmsk2008 1. Calculate a. Calculate the of the cuboid shown. b. Calculate the of the cuboid shown. 2 2 2 18 5 2 5 4 2 18 4 2 2 364 5 18 4 3 360 2. Calculate the a and b total of the cylinder shown . 2 2 2 6 4 2 4 2 2 2 34 . 251 2 6 4 2 3 63 . 301 Page 26 of 29 gagaulih@jeffery_jmsk2008 Definition of similar shape a. Similarity generalizes the study of that have the same shape but not necessarily the same size. b. Let consider two similar cones shown in figure 3.10 below. c. Assuming the two shapes is similar, then 1. 2 1 2 1 2. 2 2 2 1 2 1 3. 2 2 2 1 2 2 2 1 2 1 4. 3 2 3 1 2 1 5. 3 2 3 1 2 1 = 3 2 3 1 where: 1 height of cone 1 2 height of cone 2 1 radius of cone 1 2 radius of cone 2 1 Area of cone 1 2 Area of cone 2 1 Volume of cone 1 2 Volume of cone 2 d. Note that, these 5 formulas are also for any similar shape including cubic, cuboid cylinder, pyramid, sphere etc. h 2 h 1 r 1 r 2 Page 27 of 29 gagaulih@jeffery_jmsk2008 10. Find the perimeter and area for each following figure. a b c d 11. Find the perimeter and area for each shape, giving the answers correct to 1 decimal place. a b c d 12. Find the perimeter each shape. a b Page 28 of 29 gagaulih@jeffery_jmsk2008 4. a A circle has a radius of 15cm. Calculate the area of the circle. Show your working. b A different circle has a circumference of 120 cm. What is the radius of the circle? c Calculate the perimeter and area of this shape by giving your answer correct to the nearest cm. 5. This shape is designed using 3 semi circles. The radii of the semi circles are 3 and . a Find the area of each semi circle, in term of b If the area Find the value of a. 6. Calculate the volume and surface area of each of the following shape. a b Page 29 of 29 gagaulih@jeffery_jmsk2008 h 2 h 1 r 1 h 2 c d 7. The figure shows a wooden block that has had a hole drilled in it. The diameter of the hole is 2 cm. Calculate the of this solid, giving your answer correct to 2 decimal places. 8. The figure shows two cones which is similar. If the first cone has 10 , 4 1 1 , while the second cone has 5 . 1 2 . a. Find the height for the second cone, 2 . b. Find the area of the first cone. c. Find the volume of the second cone. 9. Calculate a The surface area with diameter is 150.2 mm. b Surface area with diameter 75.1 mm.

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