Constructions. Constructions. Curriculum Ready.

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1 onstructions onstructions urriculum eady

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3 onstructions are all about creating precise, accurate mathematical diagrams. ompass encil ule/straight edge Equip yourself with these tools and you will soon be creating cool constructions like this: entre of an Incircle: circle whose circumference just touches each side of the triangle. Give this a go! raw (or construct) an angle below that is exactly 45 o at the end point on the ray below using only a compass and a straight edge! Work through the book for a great way to do this onstructions Mathletics assport 3 Learning I SEIES 17 TI 1

4 How does it work? onstructions onstruction Terms To construct geometric shapes or properties, these terms are important and used often. Term escription icture Line segment straight line with a definite start and end point ay straight line with a definite start and no end point Line straight line which continues indefinitely in either direction Intersection point or oint of intersection The point where lines (curved or straight) cross each other Where lines cross erpendicular lines Two straight lines that cross each other at exactly 90 o Small box means 90 o erpendicular line bisector line that forms a 90 o angle with an interval and cuts it in half 0 Means same length, = ngle bisector line that divides an angle in half rc art of a circle drawn using a compass rcs of equal radii (radii is plural for radius) Two arcs drawn without changing the width of the compass oint on a line particular location on a line, usually labelled with a letter n external point point that is not located on the line or object drawn I 17 onstructions SEIES TI Mathletics assport 3 Learning

5 How does it work? onstructions Line construction: erpendicular bisector The perpendicular bisector is a line that passes through the midpoint of an interval, at a right-angle (90 o ) to it. 1 Set up Start with line segment raw two arcs, one above and one below the line segment Set the compass to more than half-way along the line segment 3 epeat step from the other end. Each new arc must cross the first pair 4 ule a line joining the intersections of the arcs Mark in the right-angle and call it M (for midpoint) M For all constructions, these thin, light construct lines must be left on the diagram. It is the working out for these types of questions. onstructions Mathletics assport 3 Learning I SEIES 17 TI 3

6 LINE NSTUTIN: EENIUL ISET * How does it work? Your Turn onstructions Line construction: erpendicular bisector raw the perpendicular bisector for the line segments or sides labelled below. Show all construct lines used. M 1.../.../ I 17 onstructions SEIES TI Mathletics assport 3 Learning

7 How does it work? onstructions Line construction: erpendicular line at a point on the line This is a line that is perpendicular to a line, passing through a specific point marked on it. 1 Set up onstruct a line perpendicular to, through point raw two arcs across the line on either side of point Set the compass to a radius that would cross the line either side of the point (when the point of the compass is at ) 3 4 Make the radius of the compass larger than before raw an arc from each of the intersection points from step, making sure they cross each other ule a line XY joining the intersection of the arcs and the point Mark in the right-angle X Y onstructions Mathletics assport 3 Learning I SEIES 17 TI 5

8 EENIUL LINE T INT N THE LINE * How does it work? Your Turn onstructions Line construction: erpendicular line at a point on the line raw a perpendicular line XY that cross the line segment/side below through the point. Show all construct lines used. 1.../.../0... LINE NSTUTIN What special geometrical line have you constructed for this hexagon? F E 3 What is special about the perpendicular to the line and the point for this construction? 6 I 17 onstructions SEIES TI Mathletics assport 3 Learning

9 How does it work? onstructions Line construction: erpendicular line through a point external to the line This is a line that is perpendicular to a given line that passes through a specific point external to the line. 1 Set up onstruct a line perpendicular to through the external point raw two separate arcs across the line Make the radius of the compass long enough to cross the line when the compass is at 3 Keep the radius of the compass the same raw an arc from each of the intersection points from step, on the other side opposite point 4 ule a line joining the intersection of the arcs and the point Mark a right-angle where the line crosses the line Make sure they cross each other onstructions Mathletics assport 3 Learning I SEIES 17 TI 7

10 EENIUL LINE THUGH INT EXTENL T THE LINE * How does it work? Your Turn onstructions Line construction: erpendicular line through a point external to the line raw a perpendicular line XY that cross the line segment/side below through the external point. Show all construct lines used. 1 3 For this construction, one arc will need to pass through the vertex. LINE NSTUTIN.../.../ I 17 onstructions SEIES TI Mathletics assport 3 Learning

11 How does it work? onstructions Line construction: arallel line through a point external to the line This is a line that is parallel to a given line that passes through a specific point external to the line. 1 Set up onstruct a line perpendicular to through the external point ule a line through and the line at an angle Set the radius of the compass to less than the distance raw a single arc through and through the line raw another similar arc with the compass at point 3 4 Set the radius of the compass to equal the distance between the two points of intersection of the first arc Move the compass to point and draw an arc across it to find point S ule a line joining points and S S S onstructions Mathletics assport 3 Learning I SEIES 17 TI 9

12 How does it work? Your Turn onstructions Line construction: arallel line through a point external to the line onstruct a line segment XY parallel to, passing through the external point. Show all construct lines used. 1 N INE LINE NSTUTIN LLEL LINE THUGH INT EXTENL T THE LINE *.../.../0...../ Use either line to construct a line parallel to JK and LM, that passes through. J K L M 3 reate a parallelogram by constructing lines parallel to the lines and, both passing through. 10 I 17 onstructions SEIES TI Mathletics assport 3 Learning

13 How does it work? Your Turn onstructions pplications of line constructions 1 Follow these steps to find the centre of the circle below. (i) onstruct a line perpendicular to the interval, passing through the circumference point. Label the other point where this new perpendicular line crosses the circle circumference E. (ii) onstruct another line perpendicular to the interval, passing through the circumference point. Label the other point where this new perpendicular line crosses the circle circumference F. (iii) Use a straight edge to draw in the intervals E and F. Where they cross is the centre of the circle. Follow these steps to join two straight parallel line segments with a smooth, continuous curve. (i) Join with a straight line and then construct a perpendicular bisector to the line. Label the point of intersection M. (ii) onstruct perpendicular bisectors to the sub intervals M and M. (iii) onstruct a line perpendicular to, down from the point to intersect with the perpendicular bisector of M. (iv) lace compass point on the new intersection point and draw an arc from to M. (v) onstruct a line perpendicular to, up from the point to intersect with the perpendicular bisector of M. (vi) lace the compass point on the new intersection point and draw an arc from to M. * WESME *.../.../0... * WESME * onstructions Mathletics assport 3 Learning I SEIES 17 TI 11

14 How does it work? onstructions ngle construction: opying an angle ngles between 0 and 180 can be copied (or duplicated) using the following construction techniques. = F E 1 Set up opy this angle (or construct a congruent angle ) raw an arc crossing both arms of angle Label the intersection points and raw a similar arc from Label the intersection point E ongruent means equal raw a ray with the start point labelled (in this case, with ) E 3 4 Set the radius of the compass to the distance From point E, draw an arc across the one from step ule a ray starting from point through point F Label the point of intersection F F F E E = FE 1 I 17 onstructions SEIES TI Mathletics assport 3 Learning

15 YING N NGLE * YING N NGLE * YING N NGLE * How does it work? Your Turn onstructions ngle construction: opying an angle onstruct copies of each of these angles. Show all construct lines used. 1 Use the exact same method when copying obtuse angles. NGLE NGL NSTUTIN TIN YI.../.../0.../ 3 For this diagram, construct E as an exact copy of What geometrical statement can you make about the rays and following the construction of the angle E? E onstructions Mathletics assport 3 Learning I SEIES 17 TI 13

16 How does it work? onstructions ngle construction: isecting an angle ngles can be divided into two, equal smaller angles using these construction techniques. 1 Set up onstruct the bisector (a line that cuts it into equal parts) of this angle Without changing the compass, draw an arc further out from point raw an arc across both rays Label the intersection points and 3 4 epeat step with the compass at point Label the point of intersection ule a ray (the bisector) from the vertex through the point Mark the equal angles with dots isector = ` if = 50o, = = 5o 14 I 17 onstructions SEIES TI Mathletics assport 3 Learning

17 How does it work? Your Turn onstructions ngle construction: isecting an angle isect these angles. Show all construct lines used. 1 onstruct the bisector for (i) and (ii) XYZ (i) (ii) Y NGLE NSTUTIN.../.../0... N NGLE * ISETING N NGLE * ISETING N NGLE * ISETING X Z Use the exact same method when bisecting obtuse angles. onstruct the bisector EG for the obtuse EF. E Name the two equal angles formed by the bisector of EF. F 3 rove using construction methods, whether or not the ray S bisects below: S Is the ray S the bisector of? onstructions Mathletics assport 3 Learning I SEIES 17 TI 15

18 How does it work? onstructions ngle construction: 60 o and 30 o angles ngles of specific sizes can be constructed using a compass and straight edge. 60o 30 o 60 o raw a large arc from point 30 o raw a large arc from point With the compass on the intersection point, draw an equal sized arc that crosses the first one With the compass on the intersection point, draw an equal sized arc that crosses the first one From the new intersection point, draw an equal sized arc across the second one raw a line from through the intersection of the two arcs Label the angle 60 o 60 o ule a line from through the new intersection of the two arcs o 16 I 17 onstructions SEIES TI Mathletics assport 3 Learning

19 How does it work? Your Turn onstructions ngle construction: 60 o and 30 o angles Show all construct lines used. 1 onstruct the following sized angles with the vertex at the start of the ray, X (i) 60 o (ii) 30 o X X (i) onstruct a 60 o angle with the vertex at the point on the line segment below. (ii) isect this new angle to split it into two 30 o angles. NSTUTIN NGLE 60 N 30 NGLES * 60 N 30 NGLES * 60 N 30 NGLES *.../.../0... / 3 (i) onstruct a 30 o angle with the vertex at point X on the line segment below. (ii) onstruct a 60 o angle with the vertex at point Y on the line segment below. (iii) Extend the two constructed arms until they meet to form a triangle at point Z. What type of triangle (ΔXYZ) has been formed by combining these angle constructions? X Y onstructions Mathletics assport 3 Learning I SEIES 17 TI 17

20 How does it work? onstructions ngle construction: 45 o and 90 o angles ngles of specific sizes can be constructed using a compass and straight edge. 45 o 45 o onstruct a perpendicular bisector to the line segment (go back to check if you forget how) 90 o With the compass at, mark a dot above the line Set the compass to the distance M and draw an arc from to the perpendicular bisector M Flip the compass around and draw a large arc from above and through the line twice Label the second intersection point M ule a line from through the intersection of the arc and perpendicular bisector Label the angle 45 o 45 o M ule a line from, through until it crosses the arc again at ( is the diameter of the circle) ule a line from to to make a right angle Mark a right angle at I 17 onstructions SEIES TI Mathletics assport 3 Learning

21 45 N 90 NGLES * 45 N 90 NGLES * 45 N 90 NGLES * How does it work? Your Turn onstructions ngle construction: 45 o and 90 o angles Show all construct lines used for these construction questions. 1 onstruct the following sized angles with the vertex at M. NSTUTIN TI NGLE.../.../0.../0 (i) 45 o (ii) 90 o M N L M (i) raw (or construct) an angle below that is exactly 45 o at the end point on the ray below. (ii) opy the angle from part (i) at to create a pair of parallel lines. emember me? 3 (i) onstruct a 90 o angle with the vertex at point X on the line below. (ii) onstruct a 45 o angle with the vertex at point Y on the line below. (iii) Extend the two constructed arms until they meet (point Z) to form a right-angled triangle. What is special about the right-angled triangle formed in this construction? X Y onstructions Mathletics assport 3 Learning I SEIES 17 TI 19

22 MINING LINE N NGLE NSTUTINS * How does it work? Your Turn onstructions ombining line and angle constructions Show all construct lines used for these trickier questions requiring combinations of construction techniques. 1 onstruct the following sized angles with the vertex at X. (i) 15 o hint: this is half of 30 o (ii).5 o hint: this is half the size of which angle? W X X Y onstruct a 75 o angle with the vertex at the point on the ray below. hint: this is 15 o less than a 90 o angle or 30 o more than a 45 o angle..../.../ (i) onstruct a perpendicular bisector to the interval XY. Label the intersection point. (ii) onstruct two 45 o angles below the interval from each end point X and Y. raw the arms until they intersect. (iii) onstruct a 30 o angle with the vertex at point X above interval XY. (iv) onstruct a 60 o angle with the vertex at point Y above interval XY. raw the arm so it intersects with the 30 o angle. (v) Set the compass to the radius X and draw a complete circle. X Y What is special about the intersection of the angles and the circle drawn? 0 I 17 onstructions SEIES TI Mathletics assport 3 Learning

23 How does it work? onstructions yclic polygon construction: egular hexagon inscribed in a circle shape inscribed inside a circle is one where all the vertices touch the circumference of a circle. These shapes are called cyclic polygons. 1 Set up raw a complete circle and label the centre Without changing the compass, place the point on the circumference and draw a small arc 3 4 Move the compass point to the arc drawn, and repeat the same process all around the circle ule a line from one arc intercept to the next to create a regular hexagon onstructions Mathletics assport 3 Learning I SEIES 17 TI 1

24 YLI LYGN NSTUTIN: EGUL HEXGN INSIE IN ILE How does it work? Your Turn onstructions yclic polygon construction: egular hexagon inscribed in a circle Show all construct lines used for these. 1 (i) onstruct a cyclic hexagon EF in the circle below using the point the first vertex. (ii) Keeping the radius of the compass the same, put your compass point at each vertex of the hexagon and draw arcs starting and finishing on the circumference to produce a flower petal pattern..../.../0... (i) onstruct a hexagon UVWXYZ in the circle with centre below using the point U as the first vertex. (ii) onstruct a 30 o angle using the side UV, with the vertex at U. Extend the newly constructed arm until it meets the circumference of the circle. (iii) onstruct a 60 o using the side WX, with the vertex at X. Extend the newly constructed arm all the way across the circle. fter the straight line UX is drawn in, what special type of triangle is formed? U I 17 onstructions SEIES TI Mathletics assport 3 Learning

25 Where does it work? onstructions Triangle construction: Equilateral triangles The basic construction techniques covered earlier can be applied to create specific shapes. Equilateral triangle (side length method) Equilateral triangle (angle method) Set the compass to the length of the interval onstruct a 60 o angle at on the line segment raw a large arc from one end () onstruct another 60 o angle at o the same from the other end of the line ule the sides from and to the point of intersection ule over the side constructions to the point of intersection to create an equilateral triangle o 60 o 60 o onstructions Mathletics assport 3 Learning I SEIES 17 TI 3

26 TINGLE NSTUTIN: EUILTEL TINGLES Where does it work? Your Turn onstructions Triangle construction: Equilateral triangles Show all construct lines used for these triangle constructions. 1 onstruct equilateral triangles on XY below using the method indicated..../.../0... (i) onstructing the angles (ii) onstructing the side lengths X Y X Y The angle method can also be used to construct isosceles triangles. onstruct an isosceles triangle using the given base, and with two equal angles of 30 o emember: Isosceles triangles have two equal sides, opposite equal angles base of isosceles triangle 3 (i) reate a rhombus JKLM by constructing equilateral triangles on either side of the line segment below. (ii) onstruct the perpendicular bisector to the diagonal JL, ensuring the line passes through K and M. (iii) Use an angle construction technique to see if the perpendicular bisector of JL also bisects JKL. J What geometric property of a rhombus has been shown by these constructions? 4 I 17 onstructions SEIES TI Mathletics assport L 3 Learning

27 Where does it work? onstructions Triangle construction: Median of a triangle median is a line which divides the area of a triangle down the middle into two equal halves. 1 Median of Δ rea of triangle 1 = rea of triangle 1 Set up onstruct the median of Δ from the side through the vertex onstruct the perpendicular bisector through 3 4 Mark the midpoint of the line with a letter () Mark the equal lengths and ule a line from to the vertex rea of Δ = rea of Δ onstructions Mathletics assport 3 Learning I SEIES 17 TI 5

28 TINGLE NSTUTIN: MEIN, ENTI N IUMENTE Where does it work? Your Turn onstructions Triangle construction: Median, centroid and circumcentre Show all construct lines used for these triangle constructions. 1 onstruct the median of the triangle below from the midpoint (M) of the side XY through the vertex W. X Name the two equal sized triangles formed by your construction. W Y The centroid () is the point where all the medians drawn from each side of a triangle intersect. Find the centroid by constructing the median lines from every side of this triangle..../.../0... ll the small triangles formed by the three median lines will have the exact same area! 6 I 17 onstructions SEIES TI Mathletics assport 3 Learning

29 Where does it work? Your Turn onstructions Triangle construction: Median, centroid and circumcentre The circumcentre of a triangle is found using the same construction methods as those used for finding the centroid of a triangle. The only difference is that instead of drawing in the median lines, we look at where the perpendicular bisectors of each side cross each other. This point is the circumcentre of the triangle. 3 Follow these steps to construct the circumcentre for ΔJKL below. (i) onstruct the perpendicular bisector for the side JK. (ii) onstruct the perpendicular bisector for the side KL. (iii) onstruct the perpendicular bisector for the side JL. (iv) Use a straight edge to extend the perpendicular bisectors to find the point they all intersect each other. Label the circumcentre (). K J L 4 onstruct the circumcentre for ΔXYZ below. For this one, the circumcentre will be outside the triangle. o the constructions and see for yourself! Y Z X onstructions Mathletics assport 3 Learning I SEIES 17 TI 7

30 Where does it work? onstructions Triangle construction: rthocentre of a triangle The orthocentre of a triangle is where the altitudes of a triangle all intersect each other. The altitude of a triangle is a line that passes through the vertex, perpendicular to the side opposite it. 1 Set up onstruct the orthocentre () of Δ below onstruct the line perpendicular to, passing through the vertex is extended (produced) to S with a dotted line to enable the following construction onstruct the line perpendicular to, passing through the vertex S roduce to T to enable next construction onstruct the line perpendicular to, passing through the vertex Label the orthocentre () S T The correct term used to extend lines is produced. The order that the line segment is named is important. The line segment is produced to S The line segment is produced to S S S 8 I 17 onstructions SEIES TI Mathletics assport 3 Learning

31 TINGLE NSTUTIN: THENTE F TINGLE Where does it work? Your Turn onstructions Triangle construction: rthocentre of a triangle Show all construct lines used for these. onstruct the orthocentre () for these triangles below: 1.../.../0... E F 3 You will need to produce the side IH to J for this one. 4 You will need to produce LM and KM. The perpendicular line for vertices K and L will join with these produced lines outside of the triangle. G K I M L H onstructions Mathletics assport 3 Learning I SEIES 17 TI 9

32 TINGLE NSTUTIN M TIME: THE EULE LINE Where does it work? Your Turn onstructions Triangle construction combo time: The Euler line Leonhard Euler (pronounced iler) was a Swiss mathematician who discovered that the orthocentre, centroid and circumcentre for any triangle are collinear. This means they all lie in a perfect straight line. retty cool! For the triangle below, construct the orthocentre, centroid and circumcentre and join them with a straight line to show Euler s line. X Y Z.../.../ I 17 onstructions SEIES TI Mathletics assport 3 Learning

33 What else can you do? onstructions ircle construction: circle that passes through three non-collinear points Non-collinear points are points that together, do not form a straight line. X Y Z 1 Set up onstruct a circle that passes through these three non-collinear points ule two line segments between two pairs of points (eg and ) onstruct the perpendicular bisector through both line segments Label their intersection point Set the radius of the compass to the distance from to any of the points, or With the compass at, draw a full circle, passing through all the points onstructions Mathletics assport 3 Learning I SEIES 17 TI 31

34 ILE THT SSES THUGH THEE NN-LLINE INTS * What else can you do? Your Turn onstructions ircle construction: circle that passes through three non-collinear points Show all construct lines used. ILE For each group of three points below, construct the circle that passes through them..../.../0... NSTUTIN 1 a b M L N ircumscribing a circle on a triangle means drawing a circle that touches each vertex of a triangle. So it is exactly the same method for construction, simply replacing the points with the vertices. ircumscribe a circle on each triangle below by following the same steps as a circle passing through three points. a b L M N 3 I 17 onstructions SEIES TI Mathletics assport 3 Learning

35 What else can you do? onstructions ircles: Tangents to a circle from an external point tangent is a line that just touches the circumference of a circle at one point. Two or more lines that pass through the same point are called concurrent lines. 1 Set up onstruct tangent lines through the external point to the circle shown onstruct the perpendicular bisector to find the midpoint (M) of the line segment ule a line from the circle () to the point () M 3 4 Set the radius of the compass to the distance M ule the tangents to the circle at and, that pass through the external point raw a circle with the compass at M Label the points it crosses the circumference and M M onstructions Mathletics assport 3 Learning I SEIES 17 TI 33

36 ILES: TNGENTS T ILE FM N EXTENL INT * What else can you do? Your Turn onstructions ircles: Tangents to a circle from an external point Show all construct lines used. 1 onstruct tangent lines that are concurrent (pass through the same point) with the given external point for the two circles below. a b G I The two circles below are the exact same size. (i) Use a straight to draw a line segment that joins the centre points. (ii) Use your construction skills to find the midpoint (M) of the line segment. (iii) onstruct tangent lines from the point M to the circle with centre. (iv) Show that these tangents are also tangents to the circle with centre by extending them..../.../ I 17 onstructions SEIES TI Mathletics assport 3 Learning

37 What else can you do? onstructions ircles: Incircle of a triangle Incircles are drawn inside a triangle with the circumference just touching each side once. 1 Set up onstruct the incircle of the triangle isect any two angles (in this case and ) Extend the bisectors until they cross each other () 3 4 onstruct a line perpendicular to the side common to both angles (), passing through the point With the compass set to the distance, draw a circle with the compass at point emember: lways leave the construct lines on your drawing Here is what the incircle of Δ looks like without the construct lines onstructions Mathletics assport 3 Learning I SEIES 17 TI 35

38 What else can you do? Your Turn onstructions Triangle constructions: Incircle of a triangle Show all construct lines used for these constructions. For Δ below: (i) onstruct the bisector for. (ii) onstruct the bisector for. Extend bisector to intersect with the bisector of and label. (iii) onstruct a line perpendicular to the side through the point. (iv) Use your compass to draw the incircle of with centre at point. ILES: INILE F TINGLE *.../.../0... ILES: INILE F TINGLE * 36 I 17 onstructions SEIES TI Mathletics assport 3 Learning

39 17 I SEIES TI 37 onstructions Mathletics assport 3 Learning heat Sheet onstructions Here is a summary of the important things to remember for constructions M Line construction: erpendicular bisector Line construction: erpendicular line at a point on the line Line construction: erpendicular line through a point external to the line Line construction: arallel line through a point external to the line ngle construction: opying an angle E F E E F

40 heat Sheet onstructions ngle construction: isecting an angle ngle construction: 60 o and 30 o angles 60 o 60 o 30 o 30 o ngle construction: 45 o and 90 o angles 45 o M M M 45 o 90 o yclic polygon construction: egular hexagon inscribed in a circle Triangle construction: Equilateral triangles Side length method ngle method 60 o 60 o 60 o 38 I 17 onstructions SEIES TI Mathletics assport 3 Learning

41 17 I SEIES TI 39 onstructions Mathletics assport 3 Learning heat Sheet onstructions Triangle construction: Median of a triangle Triangle construction: rthocentre of a triangle ircle construction: circle that passes through three non-collinear points ircles: Tangents to a circle from an external point ircles: Incircle of a triangle M M M

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