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Points and Lines
The Equation of a Line
Points and Lines in Axiomatic GeometriesPlane Shapes and AreaPlane Areas (square and rectangle, triangle, circle)PolygonsQuadrilaterals
Types of Quadrilaterals
Finding the Area of a QuadrilateralArea of a Polygon
DiagonalsAngles in PolygonsSum of Angles
Regular Polygons
Sum of Interior AnglesA point is often thought of as a dot on a page. Mathematicians describe it as a geometrical element that has a location but no size; a point has no dimensions. The position of a point in a plane or in space can be described using Cartesian co-ordinates. In Euclidean geometry, a line is a straight one-dimensional geometrical figure that has infinite length and no thickness, and that consists of an infinite number of points. The line joining the two points in Diagram 1 is actually called a line segment, since a line would extend to infinity in both directions.
Diagram 1
A line segment between two points is always the shortest path between them.
Any pair of points uniquely determines a line. Therefore, knowing the
co-ordinates of only two points on a line enables the equation of the line to be
found. In two-dimensional Cartesian co-ordinates, the general equation of a line
is
y = mx + c
where m is the gradient (slope) of the line and c is the y-value of the point
where the line crosses the y-axis (the y-intercept). If the line passes through
two points, (x1, y1) and (x2, y2),
then the gradient is defined as
m=(y2 - y1)/(x2 - x1)
For example, the line in Diagram 2, which passes through the points (0, 2) and (4, 4), has a gradient of
m=(4 - 2)/(4 - 0)=2/4=1/2
This means that for every 1 unit moved along the x-axis, 1/2 unit is moved along the y-axis.
Diagram 2
Substituting m = 1/2 into the general equation, we obtain
y= ½ x + c
Where the line crosses the y-axis, x = 0 and y = 2, so
2 = 1/2 × 0 + c
Therefore
c = 2
The equation of this particular line is, therefore
y= ½ x + c
The type of geometry usually taught in schools and used in everyday situations is called Euclidean geometry.
The Greek mathematician Euclid created a set of five axioms, or basic
statements, that he defined as being always true. Using these axioms, it is
possible to construct various geometrical theorems. The five axioms are:
1. A straight line can be drawn from any point to any other point.
2. A finite straight line segment can be extended continuously in a straight
line.
3. A circle can be described with any centre and any radius.
4. All right angles are equal to one another.
5. If a straight line (the transversal) meets two other straight lines so that
the interior angles formed on one side of the transversal add up to less than
two right angles, then the other two lines, when extended indefinitely, will
meet on that side of the transversal.
For two thousand years, these five axioms formed the primitive basis for all our
geometrical knowledge. However, there are words such as “point” and “line”
that occur in the statements of the axioms, but are not defined. We are so used
to representing geometrical figures in Cartesian space that we assume that “point”
and “line” must always mean what we usually picture them as –
respectively, a zero-dimensional dot, and a straight one-dimensional figure.
We can usefully reinterpret Euclid’s axioms – at least, the first four
axioms – by redefining “point” and “line”. Provided that the new
definitions are consistent within the framework of the axioms, we can construct
the same geometrical theorems as before, but the way we visualize geometrical
figures will be different. For example, in Lobachevskian or hyperbolic geometry,
which can be defined in terms of the reinterpreted axioms, a straight line looks
curved to a Euclidean observer.
Plane (flat) shapes are figures such as circles, squares, and triangles. Generally, a plane shape consists of lines or curves enclosing a region of a two-dimensional surface. Often, a plane shape is said to be closed. This means that it completely encloses a region of a surface. The area of a plane shape gives a measure of the region enclosed by the shape. It is also possible to find the area of all or part of the surface of a solid shape or a surface. This type of area is called the surface area. Areas are measured in square units, such as square centimetres or square metres.
There are standard formulas for finding the areas of many plane shapes.
The simplest plane area to find is that of a square: the area is simply the
length of one side, squared (multiplied by itself). For example, the area of the
square in Diagram 1 is given by
area = 3 cm
× 3 cm = (3 cm)2 = 9 cm2
The area of a rectangle is given by the length times the width. The area of the
rectangle in Diagram 1 is given by
area = 4 cm
× 2 cm = 8 cm2
Diagram 1
There are several different ways of finding the area of a
triangle. The basic formula is half the base times the height, that is,
area= ½ bh
For example, the area of the triangle in Diagram 2 is
1/2 × 4 cm
× 8 cm = 16 cm2
Diagram 2
If the triangle is not a right-angled triangle, then the height that needs to
be found is the perpendicular height from the base to the apex. If one of the
angles of the triangle is known, then the height can be found using one of the
basic trigonometric functions. For example, the height, h, of the triangle in
Diagram 3 can be found from
h/8=sin50°
So,
h = 8 ×
sin50° » 6.13 cm
Therefore, the area of the triangle in Diagram 3 is
1/2 × 10 cm × 8 × sin50° cm
»
1/2 × 10 × 6.13 cm = 30.64 cm2 (to two decimal places)
Diagram 3
This method can be generalized to give a second simple formula for the area
of any triangle, given the lengths of two sides and the angle between them. The
formula is
area= ½ ab
sinC
For example, the area of the triangle in Diagram 3 can be found using this method:
area = 1/2 ×
8 cm × 10 cm × sin50°
»30.64 cm2
A third formula for the area of a triangle can be used when the only information
known is the lengths of the sides. Attributed to the ancient Greek mathematician
Hero of Alexandria, Hero's formula states that
area=Ös(s-a)(s-b)(s-c)
where a, b, and c are the lengths of the sides of the triangle, and s is the
semiperimeter (half the length of the perimeter) of the triangle. In other words,
s= ½ (a+b+c)
For example, the semiperimeter of the triangle in Diagram 4 is given by
s= ½ (7+6+12)=½ x 25=12.5
Therefore, the area of the triangle in Diagram 4 is
area=Ö
12.5(12.5-7)(12.5-6)(12.5-12)
» 14.95 cm2
Diagram 4
The formula for finding the area of a circle uses the radius of the circle and
the constant p (pi). Many calculators have a p button that can be used for this
sort of calculation. Alternatively, the approximate value of p, 3.14, can be
used. The area of a circle is given by
area = pr2
For example, the circle in Diagram 5 has a radius of 4 cm, so the area of the circle is given by
area = p × 42 » 50.27 cm2
Diagram 5
Triangles, squares, and pentagons are all examples of polygons. A polygon is a plane figure with three or more straight sides. Regular polygons have sides of equal length and interior angles of equal size. Polygons may be classified as either convex or re-entrant. Each interior angle of a convex polygon is less than 180°. A re-entrant polygon has at least one interior angle greater than 180°.
A quadrilateral is a polygon with four sides. An alternative definition of a
quadrilateral is that it is a plane figure consisting of four distinct points
(called vertices) joined by four distinct line segments that intersect only at
the vertices.
Quadrilaterals can be convex or re-entrant. Each interior angle of a convex
quadrilateral is less than 180°. A re-entrant quadrilateral has at least one
interior angle greater than 180° (see Diagram 1). The sum of the internal
angles of a convex quadrilateral is 360° (2p rad).
Diagram 1: Convex and Re-entrant Quadrilaterals
There are a number of distinct, named varieties of quadrilateral including: a trapezium, which has two parallel sides; a parallelogram, which has two pairs of parallel sides; a rhombus, which is a parallelogram with all sides of equal length (a parallelogram that is not a rhombus is sometimes called a rhomboid); a rectangle, in which adjacent sides are perpendicular (at right angles to each other); and a square, which is a rectangle with all sides of equal length (or, equivalently, a rhombus with adjacent sides perpendicular; see Diagram 2)
Diagram 2: Types of Quadrilateral
A quadrilateral is cyclic if its vertices all lie on a circle. Opposite pairs of angles of a cyclic quadrilateral add up to 180° (see Diagram 3).
Diagram 3: A Cyclic (inscribed) Quadrilateral
The area of an irregular quadrilateral can be found by dividing it into two
triangles. The area of a trapezium is given by:
area= ½ h(a+b)
The area of a parallelogram (or rhombus) is given by the length of the base times the height:
area = b × h
The area of a rectangle and a square are given by the length times the width:
area = l × w
For a square, the length and the width are the same so the area is given by:
area = l × l = l2
(see Diagram 4).
Diagram 4: Finding the Area
There are standard formulas for finding the areas of certain polygons. Squares,
rectangles, and triangles have already been considered here, but there are also
standard formulas for finding the areas of parallelograms, rhombuses, and trapezium. In general, there are two methods for finding the area of a polygon
that does not have a standard formula.
Any polygon can be divided up into triangles, or rectangles, or a combination of
the two. The area of each of these shapes can then be found and added together
to give the area of the whole polygon. For example, the polygon in Diagram 6 can
be divided along the dotted line to give a rectangle and a triangle.
Diagram 6
The area of the rectangle is given by
area = 4 cm × 5 cm = 20 cm2
The area of the triangle can be found using Hero's formula. It is given by
s= ½ (2+3+4)=4.5
area=Ö4.5(4.5-2)(4.5-3)(4.5-4)
» 2.90 cm2
Therefore, the area of the polygon is given by
area = 20 cm2 + 2.90 cm2 = 22.90 cm2
The other way to find the area of a polygon is to add shapes that have a standard formula onto the polygon to make another shape with a standard formula. For example, the polygon in Diagram 7 can have two triangles added to it to make it into a rectangle.
Diagram 7
The area of the polygon is then given by the area of the large rectangle minus
the areas of the two small triangles. The area of the rectangle is given by
area = 4 cm × 7 cm = 28 cm2
The areas of the triangles can be found by using the half base times height formula, as follows:
area1 = 1/2 × 2 cm × 3 cm = 3 cm2
area2 = 1/2 × 2 cm × 1 cm = 1 cm2
So the area of the whole polygon is given by
area = 28 cm2 - (3 cm2 + 1 cm2)
= 28 cm2 - 4 cm2 = 24 cm2
Lines joining opposite vertices of a quadrilateral are called diagonals. Both
diagonals of a rectangle (or square) are equal in length (see Diagram 5).
Sometimes a quadrilateral is defined as four distinct points joined by six
distinct line segments and the diagonals are counted as sides.
Diagram 5: Diagonals
Polygons have two types of angles, interior angles inside the polygon, and exterior angles outside the polygon (see Diagram 1).
Diagram 1
Any interior angle and its corresponding exterior angle add up to 180°. A polygon has the same number of interior (and exterior) angles as it has sides.
The sum of the exterior angles of any polygon will always be 360°. The
interior angles of any N-sided polygon always add up to 180(N - 2)°. For
example, the sum of the interior angles of an octagon (8 sides) is
180(8 - 2)° = (180 × 6)° = 1080°
A regular polygon has sides of equal length and interior and exterior angles
of equal size. Therefore, the size of each exterior angle is given by
360°/N
and the size of each interior angle is given by
180(N-2)°/N
For example, the size of each exterior angle in a regular nonagon (9 sides) is given by
360°/9=40°
The size of each interior angle is given by
180(9-2)°/9=(180x7)°/9=140°
An alternative way of finding the size of an interior angle of a regular polygon
uses the fact that the sum of each interior angle and its corresponding exterior
angle is 180°. Once the size of the exterior angle has been found, the size of
the interior angle can be found by subtracting the size of the exterior angle
from 180°. For example, size of the interior angle of a regular nonagon can also be found by
180° - 40° = 140°
The sum of the interior angles of any triangle is 180°. This can be used to
show that, for any N-sided polygon, P, the sum of the interior angles of P, sum(P), is 180(N - 2)°.
Suppose N is at least 4 (N = 3 gives a triangle). Diagram 2 shows an irregular
decagon (10 sides) as an example.
Diagram 2
Now suppose that the polygon is divided into a triangle T and a smaller polygon Q. The dividing line must join two corners, b and d, that have a common neighbour, c, and it must not cross any other line (see Diagram 3). This division can be carried out for any polygon.
Diagram 3
The sum of the interior angles of P equals the sum of the interior angles of
Q plus the sum of the interior angles of T. But T is a triangle, so its angles
add up to 180°, that is sum(T) = 180°. Hence, sum(P) = sum(Q) + 180°.
Now, Q is a polygon with N - 1 sides. The new polygon, Q, can also be divided
into a triangle, T1, and a smaller polygon, R (see Diagram 4).
Diagram 4
The sum of the interior angles of Q equals the sum of the interior angles of
R plus the sum of the interior angles of T1. Again, the sum of the
interior angles of T1 is 180°, since T1 is a triangle.
The sum of the interior angles of P is now given by
sum(P) = sum(R) + 180° + 180°
This process can be repeated until the whole polygon has been divided into
triangles. A polygon with N sides divides up into N - 2 triangles (see Diagram 5).
Diagram 5
Each of these triangles has interior angles that add up to 180°, so the total sum of the interior angles is given by 180(N - 2)°.