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1 ANNOUNCEMENTS Lab. 6 will be conducted in the Computer Aided Graphics Instruction Lab (CAGIL) - 331 Block 3. You will be guided through the practical on the computer. No hidden lines in a sectioned view.

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2 Shapes of engineering components

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R105 AB C E G F K H D R20 96 70 56 30 46 30 3 60 8 holes 20, PCD 45 35 40 Geometric constructions

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TO DIVIDE A LINE To divide a given line AB into any number of equal parts. -Suppose the line AB is to be divided into 6 equal parts. 1.Draw a line AC of any length inclined to AB at some convenient angle (preferably between 20° and 40°). 2.Mark off six equal divisions on AC by cutting arcs of suitable radii consecutively starting from A. Number these divisions as 1, 2, 3, 4, 5 and 6. 3.Join 6 with B. 4.Draw lines through 5, 4, 3, 2 and 1 parallel to 6– B and cutting AB at points 5’, 4’, 3’, 2’ and 1’ respectively. Set-squares or drafter may be used for this purpose. The divisions 1’, 2’, 3’, 4’, 5’ divide the line AB into 6 equal parts.

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Draw a perpendicular bisector of a line AB Draw the line AB With A as center and radius greater than half AB but less than AB, draw an arc on either side of AB (shown green) With B as center and same radius, draw an arc on either side of AB (shown brown) Join the point of intersection of these arcs on either side of AB This is the perpendicular bisector 5

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It is similar to bisecting a line Bisecting an arc Line Arc 6 E BC>BE

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Drawing a perpendicular to a line at a given point AB P E D Draw the line AB With P as center and any convenient radius, draw an arc cutting AB in C (shown blue) With the same radius cut 2 equal divisions CD and DE (shown red) With same radius and centers D and E, draw arcs (green and brown) intersecting at Q PQ is the required perpendicular C Q 7

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Drawing a perpendicular to a line at a given point (alternate method) Cut arcs with any radius (r 1 ) on both sides of the point on the line AB. AB may be extended With a radius greater than r 1, draw arcs with centers C and D to intersect at Q QP is the required perpendicular 8

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Drawing a perpendicular from a point to a line From the external point P, draw arcs to cut the line AB at C and D. AB may be extended With C and D as centers and radius greater than half CD, draw arcs to intersect at E PE is the required perpendicular to AB 9

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To bisect a given angle AOB. 1.With O as centre and any convenient radius, mark arcs cutting OA and OB at C and D respectively. 2.With C and D as centers and same or any other convenient radius, mark two arcs intersecting each other at E. 3.Join OE. 4.OE is the bisector of AOB, i.e., AOE = 2 x EOB. Bisecting an angle 10

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Draw horizontal (1, 5) and vertical (3, 7) diameters which will be at 90 o to each other. Bisect the angles to get new diameters (2, 6) and (4, 8) at 45 o to the horizontal and vertical dimeters. The circle is divided into 8 equal sectors To divide a given circle into 8 equal parts 11

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Draw the two diameters 1–7 and 4–10, perpendicular to each other. With 1 as a centre and radius = R (= radius of the circle), cut two arcs at 3 and 11 on the circle. Similarly, with 4, 7 and 10 as the centres and the same radius, cut arcs on the circle respectively at 2 and 6, 5 and 9, and 8 and 12. The points 1, 2, 3, etc., give 12 equal divisions of the circle. To divide a circle into 12 equal parts 12

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With centre P and any convenient radius, mark off two arcs cutting the arc/circle at C and D. Obtain QR, the perpendicular bisector of arc CD. QR is the required normal. Draw the perpendicular ST to QR for the required tangent. To draw a normal and a tangent to an arc or circle at a point P on it 13

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Join the centre O with P and locate the midpoint M of OP. With M as a centre and radius = MO, mark an arc cutting the circle at Q. Join P with Q. PQ is the required tangent. Another tangent PQ’ can be drawn in a similar way. Tangent to a given arc AB (or a circle) from a point P outside it. 14

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Common external tangent to 2 circles Circle with radius R1 Circle with radius R2 Circle with radius R1-R2 O P A B See N. D. Bhatt pg. 88, 89 Given circles are with radii R1 and R2 and centers O and P respectively Draw a circle with radius R1-R2 and center O Draw a circle with dia. OP cutting the circle with radius R1-R2 at T Draw a line OT extended cutting the circle with radius R1 at A Draw a line PB parallel to OA with B lying on the circumference of circle with radius R2 Line AB is the required tangent T NOTE: PT is a tangent from point P to the circle with radius R1-R2 15

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Common internal tangent to 2 circles Circle with radius R1 Circle with radius R2 Circle with radius R1+R2 OP A B T See N. D. Bhatt pg. 89 Given circles are with radii R1 and R2 and centers O and P respectively Draw a circle with radius R1+R2 and center O Draw a circle with dia. OP cutting the circle with radius R1+R2 at T Draw a line OT cutting the circle with radius R1 at A Draw a line PB parallel to OA with B lying on the circumference of circle with radius R2 Line AB is the required tangent 16

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Example 4.24 To draw a line parallel to a given line AB and at a given distance R from it. Solution Refer Fig. 4.30. 1. Draw a perpendicular bisector of the line AB, cutting it at M. 2. Set off MN = R. Draw PQ perpendicular to MN at N. PQ is parallel to AB. Line parallel to another line

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Draw an arc (radius R) touching 2 given lines AB and AC are the given lines Draw a line PQ parallel to and at a distance R from AB Draw a line EF parallel to and at a distance equal to R from AC intersecting PQ at O With O as center and radius R draw the arc touching to 2 lines

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19 Draw an arc (radius R2) touching a given line and another arc CASE I AB is the given line Draw a line parallel to AB at a distance R2 With O ac center and radius R1- R2, draw an arc EF cutting the line at P With P as center and Radius R2, draw the required arc CASE II AB is the given line Draw a line parallel to AB at a distance R2 With O as center and radius R1+R2, draw an arc EF cutting the line at P With P as center and Radius R2, draw the required arc

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Finding the center of an arc Draw 2 chord of the arc (CD and EF in this case) Draw perpendicular bisectors of CD and EF intersecting each other at O. O is the required center.

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Curve (given radius) joining 2 other curves Draw arcs with radius R1 - R3 (center O) and R2 + R3 (center P) intersecting at Q. With center Q draw an arc with radius R3 joining the 2 curves. Draw arcs with radius R1 + R3 (center O) and R2 + R3 (center P) intersecting at Q. With center Q draw an arc with radius R3 joining the 2 curves. Draw arcs with radius R3 – R1 (center O) and R3 - R2 (center P) intersecting at Q. With center Q draw an arc with radius R3 joining the 2 curves.

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With any point O as centre and radius = AB, draw a circle. Starting from any point (say A) on the circle, mark off the five arcs of radius = AB consecutively cutting the circle at B, C, D, E and F. Join A, B, C, D, E and F for the required hexagon. To construct a regular hexagon of given side length Principle: The distance across opposite corners in a regular hexagon = 2 x side length AD = 2 x AB 22

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CONSTRUCTION OF A POLYGON N. D. Bhatt pg. 80 A 4 6 P B Draw side AB of specified length Draw a perpendicular BP at B such that BP = AB Draw a straight line joining A and P With B as center and radius AB draw arc AP Draw a perpendicular bisector of AB to meet the line AP at 4 and arc AP at 6 Locate point 5 as the midpoint of 4-5 A square of side AB can be inscribed in the circle with center 4 and radius A4 5 23

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Polygons of different number of sides on same construction Similarly a hexagon of side AB can be inscribed in the circle with center 6 and radius A6 Mark points 7, 8, 9 on the perpendicular bisector such that 5-6 = 6-7 = 7-8 = 8-9 and so on A heptagon of side AB can be inscribed in the circle with center 7 and radius A7 An octagon of side AB can be inscribed in the circle with center 8 and radius A8…and so on A 4 6 P B 5 7 8 24

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Drawing a pitch circle and marking the holes C Draw the pitch circle Since there are 8 holes, the angle between the lines joining their centers to the center of the pitch circle will be 45 o Divide the pitch circle into 8 parts by drawing lines from its center at 45 o to the adjacent one The points of intersection of these lines and the pitch circle are the centers of the required holes Draw the holes with specified diameter Pitch circle: circle on which lies certain features e.g. the centers of smaller circles or holes. Fig. shows 8 holes drawn on a pitch circle in a square plate Pitch circle 25

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Space and Shape Grade 9 Math.

Space and Shape Grade 9 Math.

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