Download Mathematicians have developed many different kinds of geometry

Mathematicians have developed many
different kinds of geometry, but the
geometry that we discuss here is the one
that you probably met in school. It is often
known as Euclidean geometry and, until
comparatively recent times, it was the only
topic taught in the middle school that
could properly be called mathematics.
Euclid was a Greek mathematician
(c. 330–c. 275BC ), born in Alexandria,
author of a famous treatise on geometry
known as the Elements (Stoicheia) and of
whom little else is known.
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One way in which it is possible to extend
the idea of ‘geometry’ is to consider points
and lines drawn on a surface other than a
plane, for example, on a sphere. On a
sphere there may be more than one shortest
route between two points, from the north to
the south pole for example, and this is why
we have listed properties of plane
geometry which you may never have
thought to question.
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The order of the points DY or YD does not
really matter when defining the line, but
the phrase ‘DY produced’ indicates that
you move in the direction of D to Y when
extending the line.
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Results of this kind are often known as
theorems, meaning that the result can be
deduced from the initial assumptions (or
axioms) and any previous theorems.
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Note that here we are just using the letters
A, B, C to represent the interior angle at
corresponding vertices.
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The solution to Question E4 is essentially a
proof of Pythagoras’s theorem. The
theorem is also discussed elsewhere in
FLAP. See the Glossary for details.
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You are not likely to meet often the terms
equiangular and equilateral used to
describe polygons, however regular is a
term that you should remember.
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pentagon (5)
hexagon (6)
heptagon (7)
octagon (8)
enneagon (9)
decagon (10)
hendecagon (11)
dodecagon (12)
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You should memorize these.
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Note that if the angle at one end of a side
and the angle opposite the given side are
known, the angle at the other end of the
side can be found because the three angles
must add up to 180°.
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Circles will be discussed later in this
module.
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If corresponding angles in two triangles are
equal, they will be similar triangles, i.e. the
same shape. However, this does not apply
to a figure with more than three sides, for
example, a square and a rectangle have all
four angles equal but they are not
necessarily the same shape.
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This is also true of any linear property of
the triangles, for example, the perimeters
are in the same ratio as the corresponding
sides. However, this is not the same as the
ratio of their areas.
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The triangles in Figure 10 are right-angled,
however, these results are true for any pair
of similar triangles.
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Many terms in geometry such as
hypotenuse, radius and diameter are
habitually used both to describe the
geometric properties of a line segment and
to mean the actual numerical value of the
length of the line segment. For example,
we often write ‘the diameter of a circle’
when strictly we should write ‘the length
of the diameter of a circle’.
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The number 360 is of course a good choice
since it has so many factors.
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Although it is more usual to write this as
11radian = 360°/2π (and
1° = 2π1radians/360), we have included the
dimensions for completeness.
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Imagine the triangle AED being ‘cut’ from
one side of the rectangle and stuck on the
other side.
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These statements can be proved using an
argument based on congruent triangles.
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Be careful when calculating the area of a
triangle such as DFC in Figure 17d. It is
important to distinguish between FC and
the perpendicular height of the triangle.
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Be careful not to confuse a segment,
(which is bounded by an arc and a chord)
with a sector, (which is bounded by an arc
and two radii).
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Note that Figure 13 is not drawn to scale.
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The prism in Question T10 would more
often be described as a hollow cylinder or
pipe.
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Calculus is covered elsewhere in FLAP.
See the Glossary for details.
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✧
SV̂B = 50° 4( UV̂Z and SV̂B are
vertically opposite angles and therefore
equal.)
UV̂S = BV̂Z = 130° 4( UV̂Z and UV̂S
form half a complete turn, while UV̂S and
BV̂Z are vertically opposite.)4❏
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✧
Obtuse, acute, reflex, reflex, right
angle.4❏
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The supplementary angles are 50°, 120°,
90°.
The complementary angles are 60°, 45°,
30°, 0°.4❏
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✧
(a) TÛY ,
(b) SV̂B ,
(c) TÛY and TÛQ (or TÛQ and QÛV,
or QÛV and VÛY , or VÛY and TÛY ),
(d) QÛV.4❏
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✧
The angle between AC and the ray is an
alternate angle to θ, and the angle between
the ray and CD is a corresponding angle to
ψ. It follows that the exterior angle ( φ ) is
equal to the sum of the opposite interior
angles (θ and ψ ). This result is true in
general, and worth remembering.4❏
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Since b2 = c2 − a2 it follows that
b 2 = 2.251cm2 and therefore
b = 1.51cm.4❏
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Yes. You can show this by checking that
the values satisfy a2 + b2 = c2 , or more
simply by noting that if you express the
given lengths as multiples of 31m ‘units’
the sides are of length 3 units, 4 units and
5 units.4❏
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No. We cannot be sure that UV = VZ.4❏
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Every square is a rectangle, but not all
rectangles are squares. Not every rectangle
is a rhombus, but every square is a
rhombus.4❏
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✧
(a) The sum of the interior angles in a
pentagon is (10 − 4) × 90° = 540°, since
there are five sides each interior angle in a
regular pentagon is 15 (540°) = 108°.
(b) The sum of the interior angles in a
regular hexagon is (12 − 4) × 90° = 720°
so that each interior angle is
1 (720°) = 120°.4❏
6
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No. There would be a free choice of the
length of any one side.4❏
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✧
In fact there are three. Look first at triangle
ABD. One angle ( AD̂B ) is a right angle;
one ( DÂB ) is α. So the third angle, AB̂D ,
must be (180° − 90° − α), and hence must
be equal to β. Triangle ABD is therefore
similar to triangle ABC, since it has the
same three interior angles. To emphasize
which sides are in the same ratio we can be
more precise in the order in which we
write the letters representing corresponding
vertices, and say that the triangles ADB
and ABC of Figure 12 are similar. This
means that
AD DB AB
=
=
.
AB BC AC
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Now look at triangle BDC. Here, two of
the angles are known BD̂C = 90° and
BĈD = β ; so the third angle (DB̂C) will
be equal to α, and so this triangle will be
similar to both ABC and ADB. Hence
triangles ADB, ABC and BDC are
similar.4❏
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✧
This minor segment of the circle is known
as a quadrant.4❏
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✧
2π radians = 360°,
therefore 1radian =
thus
360°
,
2π radians
☞
3π
360° 3π
radians =
×
= 270°
2
2π
2
2π radians
,
360°
2π radians 135° 3
×
= π radians
thus 135°=
1
4
360°
4❏
360° = 2π radians, therefore 1° =
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OT = OS (since both are the radii of the
circle)
OT̂P = OŜP = 90° (the angle between
radius and tangent) and OP (the
hypotenuse) is common to both triangles.
The triangles are therefore congruent by
case (d) in our test for congruence in
Subsection 3.1. It follows, therefore, that
the lengths PT and PS are equal.4❏
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The cross-sectional area is π1r12 = π(0.11m)2
= 0.01π1m2 , and the volume is therefore
(0.01π1m2 ) × (1.51m) = 0.015π1m3.4❏
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We have 4πr 2 = 101m2 and therefore
10
1m ≈ 0.8921m. Hence the volume
4π
is 43 π0r03 ≈ 2.9741m3.4❏
r=
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