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Example 4. What is the formula for the number of diagonals that can be drawn in a polygon with n sides?
Solution. Drawings of the pattern are shown in figure b. The least number of sides of a polygon is three, which is a triangle. The triangle has no diagonals.
The number of sides (ft) of the polygon is entered in a table and the number of diagonals is counted. It is recognizing the pattern in the table that enables the problem to be solved:
Number of sides (/?) 3 4 5 6 n
Number of diagonals 0 2 5 9
Pattern 3x0 4 x 1 5x2 6x3 n(n-
_____ _____ _____ ----- .. 2
From the pattern we can deduce the number of diagonals for a polygon with n sides. The number of diagonals that can be drawn in a polygon with n sides is ft (ft — 3)/2.
References: Diagonal, Difference Tables, Formula, Polygon, Variable.
Check with the entry Polygon for more details on the angle sum. A pentagon is a polygon with five sides. Figure a shows some pentagons; the last one is a regular pentagon.
The regular pentagon has five sides of equal length, and its five angles are of equal size (108°). It has five axes of symmetry, and the order of rotational symmetry is five
(see figure b). The regular pentagon is made up of five congruent isosceles triangles whose angles are 54°, 54° , and 72°.
To construct a regular pentagon, you can use a protractor to measure the 108° angles. The regular pentagon will not tessellate.
References: Axis of Symmetry, Congruent Figures, Isosceles Triangle, Polygon, Protractor, Regular Polygon, Rotational Symmetry.
The pentagram is a star-shaped figure formed by drawing all the diagonals of a regular pentagon (see figure a). The pentagram has five axes of symmetry and it has rotational symmetry of order five, like the regular pentagon.
Extending, with straight lines, the sides of a regular pentagon can also form a pentagram (see figure b).
References: Axis of Symmetry, Pentagon, Regular Polygon, Star Polygons.
When five squares are joined together edge to edge they form a pentomino. The joining of each pair of squares must be along a complete edge. Altogether there are 12 different combinations of joining together five squares, and each one is called a pentomino. Eight of the pentominoes are nets for an open box. Remember that types like the three depicted in figure a are regarded as the same pentomino.
The 12 pentominoes are drawn in figure b and the squares that are the bases of the open boxes are shaded.
Jigsaw There are 12 pentominoes each made of 5 squares. Altogether there are 5 x 12 = 60 squares. If you imagine a rectangle that is made of 60 squares, then its
sides could be 6 by 10, say. It is possible to jigsaw the 12 pentominoes to fit into this rectangle, as shown in figure c. There are many solutions to this jigsaw problem.
References: Hexomino, Nets, Polyominoes.
The abbreviation for percentage is percent, or %. This latter symbol is made up by rearranging a 1 and 00. Percentage means “out of 100.” For example, 85% means 85 out of 100, which can be expressed as the fraction 85/100, which should then be canceled down to its simplest form as a fraction, which is 17/25.
In the following examples we leam how to convert % to decimals and back again, and how to convert % to fractions and back again.
Example 1. Jacob scored 57% in a math test. Write his mark as a decimal.
Solution. The rule for converting a % to a decimal is to write it over 100, and then express this fraction as a decimal:
In a similar way,
100 = 0.06
Example 2. Of the students in Nathan’s class at school, 0.25 are left-handed. Write this as a %.
Solution. The rule for converting a decimal to a % is to multiply the decimal by 100:
0.25 x 100 = 25 0.25 = 25%
In a similar way, 1.842 when converted to a % is
1.842 x 100 = 184.2 1.842 = 184.2%
This example demonstrates that a % can be greater than 100%, but a mark in an examination can never be greater than 100%.
Example 3. About 90% of an iceberg is below the surface of the water. Write this as a fraction.
Solution. The rule for converting a % to a fraction is to first write the % over 100, and then cancel down the resulting fraction:
90% = — Which cancels down to ^
Example 4. Anne sells French bread at a profit of 115%. Write this as a fraction.
115% =--------------- Which cancels down to
= 1 — Writing the improper fraction || as a mixed number Example 5. Jacob scored 13 out of 15 in an English test. Write this as a %.
Solution. 13 out of 15 is the fraction 13/15. To convert a fraction to % we multiply
the fraction by 100, which is the same as the method for converting decimals to %:
— x 100 = 86.7 (to 1 dp) Using a calculator
13 out of 15 is a mark of 86.7%.