Definition
A tangent to a circle it a ttraight lino which touches the circle at one point.
Every curve ever drawn could have tangent« drawn to it but this chapter is concerned only with tangents to circles. These have wide applications in Engineering Drawing since the outlines of most engineering details are made up of straight lines and arcs. Wherever a straight line meets an arc, a tangent meets a circle.
Constructions
To draw a tangent to e circle from any point on the circumference (Fig. 5/1 )
2. At any point on the circumference of a circle, the tangent and the radius are perpendicular to each other. Thus, the tangent is found by constructing en angle of 90* from the point where the radius crosses the circumference.
A base geometric theorem is that the angle in a semicircle is a right angle (Fig. 5/2). This fact is made use of in many tangent constructions.
To construct a tangent from a point P to a circle, centre O (Fig. 5/3)
2. Erect a semi -circle on 0 P to cut the circle in A.
PA produced is the required tangent (OA is the radius and is perpendicular to PA since rt is the angle in a semicircle). There are. of course, two tangents to the circle from P but only one has been shown for clerity.
To conetruct a common tangent to two equal circlea (Fig. 6/4)
2. From each centre, con struct lines at 90* to the centre line. The intersection of these perpendiculars with the circles gives the points of tangency.
This tangent is often descnbed at the common extenor tangent.
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To construct the common interior (or transverse or cross) tangent to two equal circles, centres O
3. Bisect OA In B and draw a semi-circle, radius BA to cut the circle in C.
t respectively (Fig 5/6)
1. Join the centres 00,.
2. Bisect 0 O, in A and draw e semi-circle, radius AO.
3. Draw a circle, centre 0, radius R-r. to cut the semi-
t respectively (Fig 5/6)
1. Join the centres 00,.
2. Bisect 0 O, in A and draw e semi-circle, radius AO.
3. Draw a circle, centre 0, radius R-r. to cut the semi-
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To construct the common internal tangent between two unequal circle«, centres O and O, and radii R andr respectively (Fig. 5/7)
1. Join the centres 00,.
2. Bisect 0 0, in A and draw a semi-circle, radius OA.
3. Drew a circle, centre 0. radius R + r. to cut the semicircle in B.
4. Join OB. This cuts the larger circle in C.
5. Draw O.D parallel to OB. CO is the required tangent
A tangent is. by definition, a straight line. However, we do often talk of radii or curves meeting each other tengentially. We mean, of course, that the curves meet smoothly and with no change of shape or bumps. This topic, the blending of lines and curves, is discussed in Chapter 8.
2 Fig. 2 shows a centre finder, or centre square in position on a 75 mm diameter bar.
Oraw, full tin. the shape of the centre finder and the piece of round bar. Show clearly the constructions
2 Fig. 2 shows a centre finder, or centre square in position on a 75 mm diameter bar.
Oraw, full tin. the shape of the centre finder and the piece of round bar. Show clearly the constructions
Exercises 5 (All questions originally set in Imperial units) 1. A former in a jig for bending metal is shown in Fig 1
(a) Draw the former, full sue. showing in full the construction for obtaining the tangent joining the two arcs
(b) Determine, without calculation, the centres of the four equally spaced holes to be bored in the positions indicated in the figure.
Middlesex Regional Examining Board
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01 MENS IONS IN mm fig. 1
01 MENS IONS IN mm fig. 1
(b) the points of contact and the centre for the 44 mm rad at B;
(c) the points of contact and the centre for the 50 mm rad at C.
South-Cast Regional examinations Board (See Ch. 8 for information not in Ch S)
Dimensions IN ■
Dimensions IN ■
3. Rg. 3 shows the outline of two pullsy wheels connected by a belt of negligible thickness. To a scalo of 1/10 draw the figure showing the construction necessary to obtain the points of contact of the belt and pulleys.
Middlesex Regional Examining Board
4. (1 ) Draw the figure ABCP shown in Fig. 4 and construct a circle, centre 0. to pass through the points A. 8 and C.
(2) Construct a tangent to this circle touching the circle at point B
(3) Construct a tangent from the point P to touch the circle on the minor arc of the chord AC.
Southern Regional Examining Board (See Ch. 4 for information not in Ch. 5)
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5. Fig. 5 shows a metal blank. Draw the blank, full size, showing clearly the constructions for obtaining the tangents joining the arcs.
7. Fig. 7 shows the outline of a metal blank. Draw the blank, full size, showing clearly the constructions for finding exact positions of the tangents joining the arcs.
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OIMCNSIOH3 IN mm ' 8 A segment of a circle stands on a chord AB which measures 50 mm The angle in the segment is 55°. Draw the segment. Produce the chord AB to C making BC 56 mm long. From C construct a tangent to the arc of the segment. University of London School Examinations (See Ch. 2 and Ch. 4 for information not in Ch. 5) 9. A and B are two points 100 mm apart With B as centre draw a circle 75 mm diameter. From A draw two lines AC and AD which are tangential to the circle AC « 150 mm. From C construct another tangent to the circle to form a triangle ACD. Measure and state the lengths CD and AD. also angle CDA Joint Matriculation Board 10. Fig. 8 shows two circles, A and 8. and a common external tangent and a common internal tangent. Construct (e) the given circles and tangents and (b) the smaller circle which is tangential to circle B and the two given tangents. Measure and state the distance between the centres of the constructed circle and circle A. Associated Examining Boa/d (See Ch. 4 for information not in Ch. 5) |
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