The Colored Pencil Course
1. Draw the given circumference AB.
3. With centre C, and radius CA, drew a semi circle.
4. With centre B. end radius BC. drew en arc to cut the semi-circle in D.
5. From 0 draw a perpendicular to AB. to cut AB m E.
6. With centre E and radius ED draw an arc to cut AB in F. AF is the required diameter.
The rest of this chapter shows some of the constructions for finding circles drawn to satisfy certain given conditions
To find the centre of any circle (Fig. 4/4)
1. Draw any two chords.
2. Construct perpendicular bisectors to these chords to intersect in 0.
0 is the centre of the circle.
To construct a circle to pass through three given points (Fig. 4/5)
1. Draw straight lines connecting the points es shown These lines are. in feet, chords of the circle
2. Draw perpendicular bisectors through these lines to intersect in 0.
0 is the centre of a circle which passes through all three points.
To construct the Inscribed circle of any reguler polygon (in thia caae. a triangle) (Fig. 4/6) 1. Bisect any two of the interior angles to intersect in 0. (If the third angle is bisected it should also pass through 0.)
0 is the centre of the inscribed circle. This centre is called the in centre
To construct the circumecribed circle of any regular polygon (in this caae a triangle) (Fig. 4/7) 1. Perpendicularly bisect any two sides to intersect In 0. (If the third side is bisected it should also pass through
0 is the centre of the circumscribed circle. This centre is called the circumcentre.
To construct the escribed circle to any regular polygon (in this case a triaogla) (Fig. 4/8)
1. An ascribed circle is a circle which touches a side and the two adjacent sides produced Thus, the first step is to produce the ad>acent sides
2. Bisect the exterior angles thus formed to intersect in 0 0 is the centre of the ascribed cirde.
To construct a circle which pasaea through a fixed point A and touches a line at a given point B (Fig. 4/9)
1. JoinAB.
2. From B erect a perpendicular BC.
3. From A construct angle BAO similar to angle CBA to intersect the perpendicular in O.
0 is the centre of the required circle.
To construct a circle which passes through two given points. A and B. and touches a given line (Fia4/10)
1. Join AB and produce this line to D (cutting the given line in C) so thst BC - CD.
2. Construct a semi circle on AD.
3. Erect a perpendicular from C to cut the semi-circle in E.
4. MekeCF-CE. 5 From F erect a perpendicular.
6. Perpendicularly bisect AB to meet the perpendicular from F in O. O is the centre of the required circle
To construct • circle which touches two given lines and passes through a given point P. (There are two circlaa which aatiafy these conditions (Fig. 4/11)
1. If the two lines do not meet, produce them to intersect inA
2. Bisect the angle thus formed.
To construct a circle, radius R. to touch another given circle radius r, and a given line (Fig. 4/12) 1 Draw a line parallel to the given line, the distance between the lines equal to R.
2. With compass point at the centre of the given circle and radius set at R + r. dtaw an arc to cut the parallel line in O.
2. With compass point at the centre of the given circle and radius set at R + r. dtaw an arc to cut the parallel line in O.
3. From any point on the bisector draw a circle, centre B. to touch the two given lines.
4. Join PA to cut the circle in C and D.
5. Draw PO, parallel to CB end PO, parallel to DB. O, and 0, are the centres of the required circles.
To construct s circle which touches another circle and two tangents of that circle (Pig. 4/13)
1. If the tangents do not intersect produce them to intersect in A.
3. From B. the point of contact of the circle and one of its tangents, construct a perpendicular to cut the bisector in 0,. This is the centre of the given circle.
4. Join BD
5. Draw EF parallel to DB and FO, parallel to BO,.
01 is the centre of the required circle.
To construct a circle which touches another circle and two lines (Fig. 4/14)
1. Draw intersecting lines parallel to the given lines These lines. AB and AC. must be distance r. the radius of the given circle, from the given lines.
To construct a circle which touches another circle and two lines (Fig. 4/14)
1. Draw intersecting lines parallel to the given lines These lines. AB and AC. must be distance r. the radius of the given circle, from the given lines.
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To construct a circle which pasaee through two given points. P and O. end touches a given circle, centra 0 (Fig. 4/15)
Fig 4/15
To construct a circle which pasaee through two given points. P and O. end touches a given circle, centra 0 (Fig. 4/15)
2. Perpendicularly bisect PQ and. with centre somewhere on this bisector, draw a circle to pass through points P end Q and cut the given circle in A and B
3. Join AB and produce to cut PQ produced in C.
4 Construct the tangent from C to the given circle. (Join CD. bisect CO in E. compass point at E draw a radius ED to cut the circle in F).
5. From F erect a perpendicular to cut the bisector of PQ in 0.
0 is the centre of the required circle.
Fig 4/15
All of the above constructions are for finding single circles which satisfy given conditions. The rest of the constructions in this chapter are concerned with more than one circle at a time.
To draw three circles which touch each other.
given the position of their centree 0„ O, and O,
1. Oraw straight lines connecting the centres.
2. Find the centre of the triangle thus formed by bisecting two of the interior angles.
3. From this centre, drop a perpendicular to cut 0,0, in A.
4 With centre 0, and radius 0 ,A. draw the first circle
5. With centre 0» and radius OA draw the second circle.
6. With centre 0, and radius 0,C (-0,8). draw the third circle.
To draw two circles. given both their radii, within a third circle, all three circles to touch each other
1. Mark off the diameter A8 of the largest circle.
2. Mark off AO, equal to the radius of one of the other circles and draw this circle, centre 0,. to cut the diameter in C.
3- From C mark off CD equal to the radius of the third Circle.
4. Mark off BE equal to the radius of the third circle
6. With centre 0 and radius OE. draw an arc to cut the first arc in 0,.
0, is the centre of the third circle.
To draw any number of equal circlea within another circle, the circles all to be in contect (in thi« case 5) (Fig. 4/18)
1. Divide the circle mto the same number of sectors at there are proposed circles.
2. Bisect all the sectors end produce one of the bisectors to cut the circle in D.
3. From D erect a perpendicular to meet OB produced in
4. Bisect DEO to meet 00 in F.
5. F is the centre of the first circle. The other circles have the same radius and have centres on the intersections of the sector bisectors and a circle, centre 0 and radius OF
Fig 4/19
Fig 4/19
To draw a number of equal circlea within a regular polygon to touch each other and one side of the polygon (inthiscase.aseptagon) (Fig 4/19)
1. Find the centra of the polygon by bisecting two of the sides.
2. From this centre, draw lines to all of the corners.
3 This produces a number of congruent triangles. All we now need to do is to draw the inscribed circle in each of these triangles This is done by bisecting any two of the interior angles to give the centre C. 4. The circles have equal radii and their cenves lie on the intersection of a circle, radius OC and the bisectors of the seven equal angles formed by step 2
3. Bisect angles CAB and D8A to intersect in E.
4. E is the centra of the first circle. The rest can be obtained by drawing a circle, radius 06. and bisecting the seven angles formed by step 2. The intersections of this circle and these lines give the centres of the other six circles.
To draw equal circles around a regular polygon to touch each other end one side of the polygon (In this case, a septegon) (Fig. 4/20)
1. Find the centre of the polygon by bisecting two of the sides.
2. From the centre O draw lines through all of the comers and produce them.
Exercises 4
(All questions onginslly set in Imperial units)
1. Construct a regular octagon on a base line 25 mm long and draw the inscribed circle. Measure and state the diameter of this circle in mm.
North Western Secondary School Examinations Boerd (See Ch. 2 lor information not in Ch. 4)
2. Describe three circles, each one touching the other two externally, their radii being 12 mm, 18 mm and 24 mm respectively
3. No construction has been shown in Fig. 1. You are required to draw the figure full size showing all construction lines necessary to ensure the circles are tangential to their adjacent lines.
Southern Regional Examinations Board
4. Construct the triangle ABC in which the bese BC = 108 mm. the vertical angle A-70° and the altitude i$ $5 mm
D is a point on AB 34 mm from A. Describe e circle to pass through the points A and D and touch (tangential to) the line BC
Southern Universities' Joint Board (See Ch. 2 for information not in Ch 4)
5. Fig. 2 shows two touching circles placed in the comer made by two lines which are perpendicular to one another. Drew the view shown and state the diameter of the smaller circle. Your construction must show clearly the method of obtaining the centre ol the smaller circle.
6 Fig 3 shows two intersecting lines AB and BC and the position of a point P. Draw the given figure end find the centre of a circle which will pass through P and touch the lines AB and BC. Draw the circle and state its radius as accurately as possible. University of London School Exeminetions
8. Construct en isosceles triengle ABC where the included angle A - 67and AB - AC » 104 mm Draw circles of 43 mm. 37 mm and 32 mm radius using as centros A, B and C respectively. Construct the smallest circle which touches all three circles.
Measure and state the diameter of the constructed circle.
AssocietedExamining Board (See Ch. 2 for information not in Ch. 4)
9. AB and AC are two straight lines which intersect at an angle of 30*. D is a point between the two lines at perpendicular distances of 37 mm and 62 mm respectively from AB and AC. Describe the circle which touches the two converging lines and passes through point D; the centre of this circle is to lie between the points A end D Now draw two other cirdee each touching the constructed circle externally and also the converging lines. Measure and state the diameters of the constructed circles.
Oxford L oca IEseminations 10 OA and 08 are two straight lines meeting at an anglo of 30° Construct a circle of diameter 76 mm to touch these two lines and a smeller circle which will touch the two converging lines and the first circle. Also construct a third circle of diameter 64 mm which touches each of the other two circles. Oxford Local Exeminetions 11. Construct a regular octagon of side 75 mm and within this octagon describe eight equal circles each touching one side of the octagon and two adjacent circles Now draw the smallest circle which will touch all eight circles. Measure and state the diameter of this circle.
Oxford Locel Exeminetions (See Ch. 2 for information not m Ch. 4)
7. A triangle has sides 100 mm. 106 mm end 60 mm long. Draw the triangle and construct and draw the following: (a) the Inscnbed circle; (b) the drcum-scribed circle; (c) the smallest escribed circle. University of London School Exeminetions (See Ch. 2 for information not in Ch. 4)
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