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# Grade 8 - geometry investigation - Rene Rix

Module by: Pinelands High School. E-mail the author

Summary: An investigation into properties of polygons

## Grade 8: Investigation paper on geometry

### General Instructions

Attempt all questions

You may write in pencil. Write neatly.

Write all answers on the question paper.

1) Consider the following diagram. If 90°+a=180°90°+a=180°90^{circ} + a = 180^{circ}, what is the value of a?

2) Consider the diagram below. What is the value of b?

3) Consider the diagram below. What is the value of c?

Consider the regular hexagon and the list of terms in the box below. Choose a term from the list in the box to fill in each of the gaps in the statements below.

1. 1) J and K are _______ of the regula hexagon
2. 2) x is an ________ of the regular hexagon
3. 3) y is an _________
4. 4) x + y = ______

Determine the value of x in each of the following triangles (show any working that you do).

1)

2)

3)

### Section B: Interior angles of a regular polygon

Consider the regular polygon shown below.

1. 1) How many sides does this shape have?
2. 2) How many vertices (corners) does this shape have?
3. 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. 4) What is the sum of the interior angles in any triangle?
5. 5) Use the answers to 3 and 4 to determine the sum of the interior angles in the pentagon.
6. 6) Use your work thus far in this task to determine the size of each interior angle in a regular polygon.

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. 1) For the last two polygons draw all possible diagonals from point A.
2. 2) Complete this table by writing the correct answer in each block

1. 1) What do you notice about the relationship between the number of vertices and the number of sides?
2. 2) There is a relationship between the number of vertices (v) and the number of diagonals (d). Write a formula for finding the number of diagonals if you are given the number of vertices. Give your answer in the form d = ….
3. 3) There is a relationship between the number of vertices (v) and the number of triangles (t). Write a formula for finding the number of triangles if you are given the number of vertices. Give your answer in the form t = …
4. 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. 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. 6) Use the work you have done thus far to help you answer the following questions:
1. What is the size of each interior angle in a regular 20-sided polygon?
2. If the sum of the interior angles of a regular polygon is 1800, how many sides does the polygon have?
3. If a regular polygon has interior angles that are each 156, how many sides does the polygon have?

### Section C: Exterior angles of a regular polygon

Consider the regular pentagon from section B. You determined that the sum of the interior angles of a pentagon is 540°540°540^{circ}. You also determined that if it is a regular pentagon, then the magnitude of each interior angle is 108°108°108^{circ}. Now we are going to look at determining the size of each of the exterior angles of a regular pentagon.

1. 1) What is the value of a in the following diagram?
2. 2) 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?
3. 3) 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?

Consider the table below. All the polygons in the table are regular. In each polygon, the sides have been produced in order.

1. 1) In the last two polygons, produce the sides in order
2. 2) Complete this table by writing only the correct answer in each block.

Study the table that you completed in the previous task and use it to answer the following questions.

1. 1) What do you observe in regard to the sum of exterior angles for all regular polygons?
2. 2) There is a relationship between the number of vertices (v) and the size of each exterior angle (e). Write a formula for finding the size of each exterior angle if you are given the number of vertices. Write it in the form e = …
3. 3) Use the work you have done so far to help you answer the following questions:
1. What is the size of each exterior angle in a regular 16-sided polygon?
2. If a regular polygon has exterior angles that are each 20°20°20^circ , how many sides does the polygon have?

### 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.

In the regular hexagon alongside, the maximum number of diagonals from each vertex has been drawn.

1. 1) How many verices are there?
2. 2) What is the total number of diagonals?
3. 3) Consider the table below.
1. Draw in all the possible diagonals in the last two polygons in the table
2. Complete the table by filling in only the correct answers.

1. 1) Refer to the table and write a formula for finding 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. 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×vd×vd times v (the number of diagonals from each vertex multiplied by the number of vertices). Write your formula in the form m = …
3. 3) Consider all the work you have done so far in this task. Determine a formula for m (the maximum number of diagonals) that only has v (no d) in it. Write the formula in the form m = …
4. 4) Determine the maximum number of diagonals that can be drawn in a 15-sided polygon.
5. 5) If it is possible to draw a maximum of 170 diagonals in a particular polygon, how many sides does this polygon have?

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