Download G1. Constructions File

Constructions
Map of Where
You Are
Grade 8
Geometry
G1.
Constructions
G2.
Informal Geometry
G3.
Angles
G4.
Lines
G5.
Triangles
G6.
Quadrilaterals
G7.
Everything Together
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 1
Throughout these exercises, do the construction on paper first,
then on GeoGebra.
Watch the video “Line Segment”.
1.
Construct a line segment with a length of:
a)
5 cm
b)
123 mm
Watch the video “Angle with Protractor”.
2.
3.
Construct angles of the following sizes:
a)
57°
d)
312°
b)
172°
Construct a line segment AB 10 cm long.
Mark a point P somewhere on this line segment.
At P construct APˆ C  29  .
Measure BPˆ C .
c)
223°
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 2
Watch the video “Circle”
1.
Construct a circle with radius:
a)
2.
7 cm
b)
39 mm
Construct line segment AB = 73 mm.
With A as the centre, construct a circle with a diameter of 10 cm.
Construct PBˆ A  153 with PB = 80 mm.
Construct QBˆ A  134  with QB = 80 mm.
Measure PBˆ Q .
With B as its centre, construct minor arc PQ.
With B as its centre, construct a major arc RS with a radius of 40 mm.
Let the intersection points of the circle and the major arc be called X and Y.
Measure the length of XY.
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 3
Watch the video “Perpendicular Bisector”
1.
Construct the following line segments and then their perpendicular bisectors:
a)
KM = 7,3 cm
b)
PQ = 137,5 mm
c)
AZ = 3,95cm
c)
WTˆP  34 
Watch the video “Angle Bisector”
2.
Construct each of the following angles and then their angle bisectors:
a)
WBˆ H  79
b)
BHˆ S  156 
d)
KLˆ M  256 
e)
XYˆZ  327 
Watch the videos “Angles without Protractor 60° 30°” and “Angles without Protractor 90° 45°”
3.
Without using a protractor, construct angles with the following sizes:
a)
60°
b)
30°
c)
90°
d)
45°
e)
105°
f)
15°
g)
22,5°
h)
26,25°
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 4
Watch the video “Triangle SSS”.
1.
Construct triangles as follows:
a)
ΔABC with a = 12 cm, b = 9 cm and c = 7 cm.
Measure the three angles of the triangle.
If you are accurate they should be 36°, 48° and 96°.
b)
ΔDEF with d = 52 mm, e = 83 mm and f = 67 mm.
Measure the three angles of the triangle.
If you are accurate they should be 39°, 54° and 87°.
c)
ΔKMP with k = 4 cm, p = 3 cm and m = 9 cm.
Watch the video “Triangle ASA”.
2.
Construct triangles as follows:
a)
ΔKLM with m = 11 cm, Kˆ  58 and Lˆ  32  .
Measure the other two sides of the triangle.
If you are accurate they should be 5,8 cm and 10,5 cm.
If you are accurate the third angle should measure 90°.
b)
ΔPQR with p = 50 mm, Qˆ  123 and Rˆ  21 .
Measure the other two sides of the triangle.
If you are accurate they should be 30,5 mm and 71,3 mm.
If you are accurate the third angle should measure 36°.
Watch the video “Triangle SAS”.
3.
Construct triangles as follows:
a)
ΔTVW with t = 7 cm, w = 8,5 cm and TVˆW  56 .
Measure the third side of the triangle.
If you are accurate it should be 7,4 cm.
b)
ΔXYZ with y = 83 mm, z = 27 mm and Xˆ  99 .
Measure the third side of the triangle.
If you are accurate it should be 91,2 mm.
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 5
Watch the video “Perpendicular from a Point”.
1.
Construct ΔABC with a = 7 cm, b = 6 cm and c = 5 cm.
From A drop a perpendicular to BC. (This is called the altitude or height of the triangle.)
Measure the length of the perpendicular from A to where it cuts BC.
If you are accurate this length should be 4,2 cm.
What is the area of the triangle?
2.
Construct ΔDEF with d = 63 mm, e = 82 mm and EFˆD  20 .
From D drop a perpendicular to FE (extended).
Measure the length of the perpendicular from D to where it cuts the extension of FE.
If you are accurate this length should be 28 mm.
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 6
Watch the video “Parallel Lines”.
1.
Construct a line segment AB = 15 cm.
Parallel to AB draw CD = 10 cm
Watch the video “Parallel Line through Point”.
2.
Construct ΔXYZ with x = 16 cm, Yˆ  45 and Zˆ  75 .
Draw a line through X parallel to YZ.
3.
Construct line segment PQ = 6 cm.
Construct a circle with centre P and radius 4 cm.
Construct a circle with centre Q and radius 5 cm.
Let one of the points of intersection of the two circles be R.
Construct a line parallel to PQ that passes through R.
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 7
1.
Construct ΔABC with a = 10 cm, b = 8 cm and c = 9 cm.
Construct the perpendicular bisector of each of the sides.
If you have worked accurately these should all pass through the same point – call it D.
Now construct a circle with centre D and radius AD.
If you have worked accurately the circle should also pass through B and C.
2.
Construct ΔABC with a = 10 cm, b = 8 cm and c = 9 cm.
Construct the angle bisector of each of the angles.
If you have worked accurately these should all pass through the same point – call it E.
Drop a perpendicular from E onto any one of the sides of the triangle.
Where it cuts the sides, label a point F.
Now construct a circle with centre E and radius EF.
If you have worked accurately the other two sides of the triangle should be tangents to the circle.
3.
Construct ΔABC with a = 10 cm, b = 8 cm and c = 9 cm.
Construct the three altitudes of the triangle.
If you have worked accurately these should all pass through the same point – call it G.
4.
Construct ΔABC with a = 10 cm, b = 8 cm and c = 9 cm.
Construct the three medians of the triangle.
(A median is a line from the vertex of a triangle to the midpoint of the opposite side.)
If you have worked accurately these should all pass through the same point – call it H.
Take one median and measure the distance along the median from H to the vertex
Now measure the distance from H to the midpoint of the side.
If you have worked accurately H should be twice as far from the vertex as from the midpoint.
8Maths4Tab
Module G1: Constructions
© 2015 Neil Eddy
Exercise 8
1.
ˆ C  120  .
Construct parallelogram ABCD with DC = 10 cm, AD = 5 cm and AD
2.
Construct equilateral triangle EFG with EF = 74 mm.
3.
Construct ΔKLM with KL = 8 cm, LM = 10 cm and KLˆ M  113 .
Construct the altitudes from each of the vertices.
4.
Construct rhombus PQRS with PQ = 6 cm and SPˆ Q  54  .
Construct diagonals SQ and PR. Label the point at which they intersect T.
Measure STˆR , SPˆ R , RPˆ Q PSˆQ and QSˆR .
5.
Construct kite WXYZ with WX = 4 cm, XY = 7 cm and ZWˆX  82  .
Construct diagonals WY and XZ.
Measure all angles.
6.
Construct a circle with diameter 14 cm.
Choose any 4 points on the circumference and label them A, B, C and D.
Connect these points to form a quadrilateral.
Construct the perpendicular bisector of each side of ABCD.
What do you notice?
Draw any quadrilateral without first drawing the circle and see if the same thing holds true.
7.
Draw any quadrilateral EFGH. (Try to make it as irregular as possible – a surprisingly difficult task!)
Construct the midpoint of each side of EFGH.
Connect these midpoints to form another quadrilateral.
What do you notice about your new quadrilateral?
8.
Draw a circle.
Construct the diameter of the circle and call it MN.
Choose any point on the circumference of the circle and call it P.
Connect MP and NP.
Measure MPˆ N .
9.
Draw a circle with radius 8 cm and centre O..
Choose any point on the circumference and call it R.
Draw OR.
Now construct POˆ R  60 with P on the circumference of the circle.
Repeat this by constructing 4 more radii at intervals of 60°.
Connect up the 6 points on the circumference to form a regular hexagon.
10.
Construct a regular octagon using a similar method.
11.
Construct a regular pentagon using a similar method.
12.
Construct a triangle with sides 15 cm, 12 cm and 8 cm.
Make 3 copies of this and construct the following points – one on each diagram:
the incentre
the orthocentre
the circumcentre
the centroid