Download Geometry basics: Polygons [NCS]

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Geometry basics: Polygons [NCS]
Free High School Science Texts Project
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Creative Commons Attribution License 3.0
1 Other polygons
There are many other polygons, some of which are given in the table below.
Sides
5
6
7
8
10
15
Name
pentagon
hexagon
heptagon
octagon
decagon
pentadecagon
Table 1: Table of some polygons and their number of sides.
Figure 1:
Examples of other polygons.
2 Extra
2.1 Angles of Regular Polygons
You can calculate the size of the interior angle of a regular polygon by using:
^
A=
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n−2
× 180◦
n
(1)
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^
where n is the number of sides and A is any angle.
2.2 Areas of Polygons
1. Area of triangle: 12 × base × perpendicular height
Figure 2
2. Area of trapezium: 21 × (sum of k (parallel) sides) × perpendicular height
Figure 3
3. Area of parallelogram and rhombus: base × perpendicular height
Figure 4
4. Area of rectangle: length × breadth
Figure 5
5. Area of square: length of side × length of side
Figure 6
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6. Area of circle: π x radius2
Figure 7
Khan Academy video on area and perimeter
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Figure 8
Khan Academy video on area of a circle
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Figure 9
2.2.1 Polygons
1. For each case below, say whether the statement is true or false. For false statements, give a counterexample to prove it:
a. All squares are rectangles
b. All rectangles are squares
c. All pentagons are similar
d. All equilateral triangles are similar
e. All pentagons are congruent
f. All equilateral triangles are congruent
Click here for the solution1
2. Find the areas of each of the given gures - remember area is measured in square units (cm2 , m2 ,
mm2 ).
Figure 10
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Click here for the solution2
3 Summary
• Make sure you know what: quadrilaterals, vertices, sides, angles, parallel lines, perpendicular lines,diagonals,
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bisectors and transversals mean.
Similarities and dierences between quadrilaterals
Properties of triangles and quadrilaterals
Congruency of triangles
Classication of angles into acute, right, obtuse, straight, reex or revolution
Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle
Angles:
· Acute angle: An angle 0 and 90
· Right angle: An angle measuring 90
· Obtuse angle: An angle 90 and 180
· Straight angle: An angle measuring 180◦
· Reex angle: An angle 180 and 360
· Revolution: An angle measuring 360
Angle properties and names
Equilateral, isoceles, right-angled, scalene triangles
Triangles angles = 180
Congruent and similar triangles
Pythagoras
Trapezium, parm, rectangle, square, rhombus, kite and properties
Areas of particular gures
4 Exercises
1. Find all the pairs of parallel lines in the following gures, giving reasons in each case.
a.
Figure 11
b.
Figure 12
c.
Figure 13
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2. Find angles a, b, c and d in each case, giving reasons.
a.
Figure 14
b.
Figure 15
c.
Figure 16
Click here for the solution4
3. Which of the following claims are true? Give a counter-example for those that are incorrect.
a. All equilateral triangles are similar.
b. All regular quadrilaterals are similar.
c. In any [U+25B5]ABC with ∠ABC = 90◦ we have AB 3 + BC 3 = CA3 .
d. All right-angled isosceles triangles with perimeter 10 cm are congruent.
e. All rectangles with the same area are similar.
Click here for the solution5
4. Say which of the following pairs of triangles are congruent with reasons.
a.
Figure 17
b.
Figure 18
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c.
Figure 19
d.
Figure 20
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5. For each pair of gures state whether they are similar or not. Give reasons.
Figure 21
Click here for the solution7
4.1 Challenge Problem
1. Using the gure below, show that the sum of the three angles in a triangle is 180◦ . Line DE is parallel
to BC .
Figure 22
Click here for the solution8
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