Download Grade 8 - geometry investigation - Rene Rix

OpenStax-CNX module: m35699
1
Grade 8 - geometry investigation ∗
Rene Rix
Pinelands High School
This work is produced by OpenStax-CNX and licensed under the
†
Creative Commons Attribution License 3.0
Abstract
An investigation into properties of polygons
1 Grade 8: Investigation paper on geometry
1.1 General Instructions
Attempt all questions
You may write in pencil. Write neatly.
Write all answers on the question paper.
1.2 Section A: Introductory tasks
1.2.1 Task one:
◦
◦
1) Consider the following diagram. If 90 + a = 180 , what is the value of a?
Figure 1
∗ Version
1.1: Sep 29, 2010 9:29 am -0500
† http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
2
2) Consider the diagram below. What is the value of b?
Figure 2
3) Consider the diagram below. What is the value of c?
Figure 3
1.2.2 Task two:
Consider the regular hexagon and the list of terms in the box below. Choose a term from the list in the box
to ll in each of the gaps in the statements below.
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
3
Figure 4
1)
2)
3)
4)
J and K are _______ of the regula hexagon
x is an ________ of the regular hexagon
y is an _________
x + y = ______
1.2.3 Task three:
Determine the value of x in each of the following triangles (show any working that you do).
Figure 5
1)
2)
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
4
Figure 6
3)
Figure 7
1.3 Section B: Interior angles of a regular polygon
1.3.1 Task one:
Consider the regular polygon shown below.
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
5
1)
Figure 8
How many sides does this shape have?
2) How many vertices (corners) does this shape have?
3) If we choose one vertex (A in the diagram) and draw all the possible diagonals from this vertex, we
see that we can draw a maximum of two diagonals, as shown. How many triangles are created within
the pentagon?
4)
Figure 9
What is the sum of the interior angles in any triangle?
5) Use the answers to 3 and 4 to determine the sum of the interior angles in the pentagon.
6) Use your work thus far in this task to determine the size of each interior angle in a regular polygon.
1.3.2 Task 2:
Consider the table below. All the polygons in the table are regular. In each polygon, diagonals have been
drawn from one vertex to all the other vertices, to divide the polygon into triangles.
1) For the last two polygons draw all possible diagonals from point A.
2) Complete this table by writing the correct answer in each block
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
6
Figure 10
1.3.3 Task three:
Use the table from task two to answer the following questions.
1) What do you notice about the relationship between the number of vertices and the number of sides?
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
7
2) There is a relationship between the number of vertices (v) and the number of diagonals (d). Write a
formula for nding the number of diagonals if you are given the number of vertices. Give your answer
in the form d = . . ..
3) There is a relationship between the number of vertices (v) and the number of triangles (t). Write a
formula for nding the number of triangles if you are given the number of vertices. Give your answer
in the form t = . . .
4) Use your formula in question 3 to help you write a formula for the sum of the interior angles (s) of any
polygon if you are given the number of vertices (v). Give your answer in the form s = . . .
5) Use your previous work in this task, together with the table in the previous task to write a formula
for determining the size of each interior angle (a) of a regular polygon if you are given the number of
vertices. Give your answer in the form a = . . .
6) Use the work you have done thus far to help you answer the following questions:
a. What is the size of each interior angle in a regular 20-sided polygon?
b. If the sum of the interior angles of a regular polygon is 1800, how many sides does the polygon
have?
c. If a regular polygon has interior angles that are each 156, how many sides does the polygon have?
1.4 Section C: Exterior angles of a regular polygon
1.4.1 Task one:
Consider the regular
pentagon from section B. You determined that the sum of the interior angles of a
◦
pentagon is ◦540 . You also determined that if it is a regular pentagon, then the magnitude of each interior
angle is 108 . Now we are going to look at determining the size of each of the exterior angles of a regular
pentagon.
1) What is the value of a in the following diagram?
2)
Figure 11
Consider producing (extending) each of the sides of a pentagon in order, as shown alongside. How
many exterior angles will be created if the sides of a pentagon are produced in order?
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
8
3)
Figure 12
All the exterior angles of a regular pentagon are the same size. What is the sum of the exterior angles
of the pentagon shown below, if the sides are produced in order?
Figure 13
1.4.2 Task two
Consider the table below. All the polygons in the table are regular. In each polygon, the sides have been
produced in order.
1) In the last two polygons, produce the sides in order
2) Complete this table by writing only the correct answer in each block.
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
9
Figure 14
1.4.3 Task three:
Study the table that you completed in the previous task and use it to answer the following questions.
1) What do you observe in regard to the sum of exterior angles for all regular polygons?
2) There is a relationship between the number of vertices (v) and the size of each exterior angle (e). Write
a formula for nding the size of each exterior angle if you are given the number of vertices. Write it in
the form e = . . .
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
10
3) Use the work you have done so far to help you answer the following questions:
a. What is the size of each exterior angle in a regular 16-sided
polygon?
◦
b. If a regular polygon has exterior angles that are each 20 , how many sides does the polygon have?
1.5 Section D: Diagonals from each vertex
In section B we looked at the maximum number of diagonals from one vertex that could divide a regular
polygon into triangles. In this section we will look at drawing all the possible diagonals within a polygon.
1.5.1 Task one:
In the regular hexagon alongside, the maximum number of diagonals from each vertex has been drawn.
Figure 15
1) How many verices are there?
2) What is the total number of diagonals?
3) Consider the table below.
a. Draw in all the possible diagonals in the last two polygons in the table
b. Complete the table by lling in only the correct answers.
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
11
Figure 16
1) Refer to the table and write a formula for nding the number of diagonals (d) from each vertex if you
are given the number of vertices (v). Write your formula in the form d = . . .
2) Use the table to help you write a formula that shows the relationship between m (the maximum number
of diagonals that may be drawn in the polygon) and d × v (the number of diagonals from each vertex
multiplied by the number of vertices). Write your formula in the form m = . . .
3) Consider all the work you have done so far in this task. Determine a formula for m (the maximum
http://cnx.org/content/m35699/1.1/
OpenStax-CNX module: m35699
12
number of diagonals) that only has v (no d) in it. Write the formula in the form m = . . .
4) Determine the maximum number of diagonals that can be drawn in a 15-sided polygon.
5) If it is possible to draw a maximum of 170 diagonals in a particular polygon, how many sides does this
polygon have?
http://cnx.org/content/m35699/1.1/