1 Construct the bisector of an angle Draw

1. Construct the bisector of an angle. • • • Draw an Acute AOB. Using the vertex O as centre draw an arc to meet the arms of the angle at X and Y. Using X as centre and the same radius draw a new arc. Using Y as centre and the same radius draw an overlapping arc. Mark the point where the arcs meet. The bisector is the line from O to this point. A X O x x y B Miss D Brennan 6 th year Constructions

2. Construct the perpendicular bisector of a line segment • • Draw the line segment Using A as centre and a radius greater than half |AB| draw an arc. Using B as centre and the same radius draw another arc. Draw a line through the points where the arcs cross. A B Miss D Brennan 6 th year Constructions

3. Line perpendicular to a given line l, passing through a given point not on l. • • Draw the line l. Mark off a point P which is not on l. ( P can be above or below l ) Using P as centre and the same radius draw two arcs to cut l at R and S. Using R as centre draw an arc on the opposite side of l to P. Using S as centre and the same radius draw another arc. Mark of the point where these arcs meet. Labelled here as T. Join T to P. P l S R T Miss D Brennan 6 th year Constructions

4. • • Line perpendicular to a given line l, passing through a given point on l. Draw the line l. Mark off a point P which on l. Using P as centre and the same radius draw two arcs to cut l at A and B. Double the size of your radius Using A as centre draw an arc on any side of l. Using B as centre and the same radius draw another arc on the same side of l. Mark of the point where these arcs meet. Labelled here as C. Join C to P and extend. l P A B C Miss D Brennan 6 th year Constructions

5. Line parallel to a given line, through given point. (Set Square Method) • • • Draw the line l. Mark off a point P which is not on l. ( P can be above or below l ) Draw a perpendicular line from P on to l. Slide the Set Square from the line l up to the point P. Draw a line from P along the base of the Set Square and extend and label as k. k P l Miss D Brennan 6 th year Constructions

5. Line parallel to a given line, through given point • • Draw a line l. Mark off a point P which is not on l. ( P can be above or below l ) Draw a line through P to cut the line l at any angle. Label the intersection A Using A as centre and a radius less than |AP| draw an arc. Add labels C and B Using P as centre and the same radius draw another arc to cut AP at R. Use the compass to measure the arc at CB. Using this as the radius and R as centre draw and arc to cut the outer arc at E. Draw a line through P and E and label as k. R k P E C l A B Miss D Brennan 6 th year Constructions

6. Divide the line segment [AB] into three equal parts • • • Draw the line segment [AB]. Through A draw a line at an acute angle to [AB]. On this line use circle arcs of the same radius to mark off three segments of equal length [AR], [RS] and [ST]. Join T to B. Through S and R draw line segments parallel to [TB] to meet [AB] at D and C. Now |AC|=|CD|=|DB| A C D B R S T Miss D Brennan 6 th year Constructions

8. Line segment of a given length on a given ray. • • • Draw a line segment |AB| of required length. (8 cm in this case). Mark off a point P on ray l. Using P as centre and a radius equal to |AB| draw an arc to cut l at C. Join P to C |PC| will now be 8 cm A B l P 8 cm Miss D Brennan 6 th year Constructions C

9. Angle of a given number of degrees with a given ray as one arm. Example to draw an angle of 45 o • • • Draw the line l Mark a point A on l. Place the centre of the protractor on the point A. Mark a point B on the circumference of the protractor at the angle required 45 o in this case. Remove the protractor and join A to B. This is the required angle. B l 45 o A Miss D Brennan 6 th year Constructions

10. Example: Construct the triangle ABC so that |AB| = 8 cm, |AC| = 6 cm and |BC| = 7 cm. • • • Draw a line segment 8 cm in length. Label the end points and mark the length. Using a compass with A as centre draw an arc of length 6 cm. With B as centre draw an arc of 7 cm. Mark their point of intersection C. Join A to C and B to C. C 7 cm 6 cm A 8 cm Miss D Brennan 6 th year Constructions B

11. Example: Construct a triangle ABC where |AB| = 12 cm, |<BAC| = 65 o and |AC| = 9 cm. • Draw a line segment 12 cm in length. Name the points and mark the length. • Use a protractor to draw a line at 65 o to |AB|. • Use a compass with A as centre and 9 cm radius to draw an arc on this line. • Mark the point of intersection C. • Join C to B to complete triangle. 9 c m C 65 o A 12 cm Miss D Brennan 6 th year Constructions B

12. Example: Construct the triangle PQR where |QR|=8 cm, |<PQR|=52 o and |<PRQ|=46 o • Draw a line segment [QR] 8 cm in length. Name the points and mark the length. • At Q using a protractor mark and draw an angle of 52 o • At R using a protractor mark and draw an angle of 46 o • Mark the point of intersection of the two angles. • This is the point P. P 46 o 52 o Q 8 cm Miss D Brennan 6 th year Constructions R

13. Example: Construct a right-angled triangle ABC, given |AB|= 6 cm, |<BAC|=90 o and |BC|=10 cm. • Draw a line segment [AB] 6 cm in length. Name the points and mark the length. • At A, using a protractor mark and draw an angle of 90 o. • Use a compass with B as centre and 10 cm radius to draw an arc on this line. • Mark the point of intersection C. Join C to B to complete triangle. • C 10 cm A 6 cm Miss D Brennan 6 th year Constructions B

14. Example: Construct a right-angled triangle PQR, such that, |QR| = 7 cm and |<QRP| = 30 o • Draw a line segment [QR] 7 cm in length. Name the points and mark the length. • At Q using a protractor mark and draw an angle of 90 o • At R using a protractor mark and draw an angle of 30 o • Extend the lines from Q and R until they meet. This is the vertex P. P 30 o 90 o Q 7 cm Miss D Brennan 6 th year Constructions R

15. Example: Construct a rectangle ABCD, with sides 6 cm and 8 cm. (Protractor) • Draw a line segment [AB] 8 cm in length. Name the points and mark the length. • At A using a protractor mark and draw an angle of 90 o • At B using a protractor mark and draw an angle of 90 o • Measure lengths of 6 cm along these perpendicular lines. . • Label these points C and D. • Join C to D. D C 6 cm 90 o A 8 cm Miss D Brennan 6 th year Constructions B

20. Example: Construct a parallelogram ABCD where |AB| = 12 cm, |<BAC| = 65 o and |AC| = 9 cm. • Draw a line segment 12 cm in length. Name the points and mark the length. • Use a protractor to draw a line at 65 o to |AB|. • Use a compass with A as centre and 9 cm radius to draw an arc on this line. • Mark the point of intersection C. • Draw a line through B parallel to AC and mark off a line segment on this of 9 cm. Call this D. D C 9 c m • Join C to D to complete parallelogram 65 o A 12 cm 3 rd year Miss D Brennan Constructions B