1. CHAPTER 9 PLANE FIGURES
A triangle is one of the basic shapes in geometry: a
polygon with three corners or vertices and three
sides or edges which are line segments. A triangle
with vertices A, B, and C is denoted .
In Euclidean geometry any three points, when non-
collinear, determine a unique triangle and a unique
plane (i.e. a two-dimensional Euclidean space).
All triangles have 3 sides and 3 angles which always
add up to 180°.
The Triangle Inequality Theorem states that:
Any side of a triangle must be:
less than the sum of the other 2 sides and
greater than the difference of the other 2 sides.
CLASSIFICATION OF TRAINGLES
Triangles are classified in 2 ways-
1) By the number of equal sides they have:
• scalene - all 3 sides have different lengths
• isosceles - 2 sides have equal lengths
• equilateral - all 3 sides are equal
2) By the types of angles they have:
• acute triangle - all 3 angles are acute (less than
• right triangle - has one right angle (a right angle =
• obtuse triangle - has one obtuse angle (an obtuse
angle is greater than 90° and less than 180°).
Relationship Between Sides of a Triangle
Here is a typical question: Two sides of a triangle
measure 2cm and 5cm. What is a possible
measurement of the third side? Sometimes you'll be
given multiple choice, and sometimes you'll have to
represent the possible range of answers. Look at the
diagram at left.
In the upper picture, I've arranged the two sides so
that they are separated quite far apart. Notice that
the missing side cannot possibly be larger than 7,
which is the sum of the two sides. If it was, then it
would not be able to connect to the other two sides. It
can't be equal to 7 either. If it was, then the two other
sides would have to be lying flat along a straight line,
and of course we then wouldn't have a triangle. They
have to be angled at least slightly. This means that x
Now let's look at the lower picture. I've arranged the
sides so that they are quite close together, forming a
small angle. Notice that x must be greater than 3,
which is the difference of the two sides. If it was less
than 3, it would be too short to connect. If it was
exactly 3, then the sides of 2 and 5 would actually be
on top of each other, and that wouldn't be a triangle
anymore. The sides of 2 and 5 must be angled at
least slightly. This means that x > 3.
We can say that 3 < x < 7. This means that x must be
between 3 and 7, but not equal to either. Any answer
within this range will be correct.
In Euclidean plane geometry, a quadrilateral is a
polygon with four sides (or edges) and four vertices or
corners. Sometimes, the term quadrangle is used, by
analogy with triangle, and sometimes tetragon for
consistency with pentagon (5-sided), hexagon (6-
sided) and so on.
The origin of the word "quadrilateral" is the two Latin
words quadri, a variant of four, and latus, meaning
Quadrilaterals are simple (not self-intersecting) or
complex (self-intersecting), also called crossed.
Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar)
quadrilateral ABCD add up to 360 degrees of arc, that
This is a special case of the n-gon interior angle sum
formula (n − 2) × 180°. In a crossed quadrilateral, the
four interior angles on either side of the crossing add
up to 720°.
All convex quadrilaterals tile the plane by repeated
rotation around the midpoints of their edges.
A convex polygon is a simple polygon whose interior
is a convex set. The following properties of a simple
polygon are both equivalent to convexity:
Every internal angle is less than or equal to
Every line segment between two vertices
remains inside or on the boundary of the
A simple polygon is strictly convex if every internal
angle is strictly less than 180 degrees. Equivalently, a
polygon is strictly convex if every line segment
between two nonadjacent vertices of the polygon is
strictly interior to the polygon except at its endpoints.
Every nondegenerate triangle is strictly convex.
A circle is a simple shape of Euclidean geometry that
is the set of all points in a plane that are at a given
distance from a given point, the centre. The distance
between any of the points and the centre is called the
radius. It can also be defined as the locus of a point
equidistant from a fixed point.
A circle is a simple closed curve which divides the
plane into two regions: an interior and an exterior. In
everyday use, the term "circle" may be used
interchangeably to refer to either the boundary of the
figure, or to the whole figure including its interior; in
strict technical usage, the circle is the former and the
latter is called a disk.
A circle can be defined as the curve traced out by a
point that moves so that its distance from a given
point is constant.
A circle may also be defined as a special ellipse in
which the two foci are coincident and the eccentricity
2. Arcs and Chords
In Figure 1, circle O has radii OA, OB, OC and OD If
chords AB and CD are of equal length, it can be
shown that Δ AOB ≅ Δ DOC. This would make m ∠1
= m ∠2, which in turn would make m = m .
This is stated as a theorem.
Figure 1 A circle with four radii and two chords
Theorem 78: In a circle, if two chords are equal in
measure, then their corresponding minor arcs are
equal in measure.
The converse of this theorem is also true.
Theorem 79: In a circle, if two minor arcs are equal in
measure, then their corresponding chords are equal
Example 1: Use Figure 2 to determine the following.
(a) If AB = CD, and = 60°, find m CD. (b)
If m = and EF = 8, find GH.
Figure 2 The relationship between equality of the
measures of (nondiameter) chords and equality of the
measures of their corresponding minor arcs.
m = 60° (Theorem 78)
GH = 8 (Theorem 79)
Some additional theorems about chords in a circle are
presented below without explanation. These
theorems can be used to solve many types of
Theorem 80: If a diameter is perpendicular to a chord,
then it bisects the chord and its arcs.
In Figure 3, UT, diameter QS is perpendicular to
chord QS By Theorem 80, QR = RS, m
= m , and m = m .
Figure 3 A diameter that is perpendicular to a chord.
Theorem 81: In a circle, if two chords are equal in
measure, then they are equidistant from the center.
In Figure 4, if AB = CD, then by Theorem 81, OX =
Figure 4 In a circle, the relationship between two
chords being equal in measure and being equidistant
from the center.
Theorem 82: In a circle, if two chords are equidistant
from the center of a circle, then the two chords are
equal in measure.
In Figure 5, if OX = OY, then by Theorem 82, AB =
Example 2: Use Figure to find x.
Figure 5 A circle with two minor arcs equal in
Example 3: Use Figure 6, in which m =
115°, m = 115°, and BD = 10, to find AC.
Figure 6 A circle with two minor arcs equal in
Example 4: Use Figure 7, in which AB = 10, OA = 13,
and m ∠ AOB = 55°, to find OM, m and m .
Figure 7 A circle with a diameter perpendicular to a
So, ST ⊥ AB, and ST is a diameter. Theorem 80 says
that AM = BM. Since AB = 10, then AM = 5. Now
consider right triangle AMO. Since OA = 13 and AM =
5, OM can be found by using the Pythagorean