Download Euclidian Geometry Grade 10 to 12 (CAPS)

Euclidian Geometry
Grade 10 to 12
(CAPS)
Compiled by Marlene Malan
[email protected]
Prepared by
Marlene Malan
CAPS DOCUMENT (Paper 2)
Grade 10
(a) Revise basic results
established in earlier
grades.
(b) Investigate line segments
joining the midpoints of two
sides of a triangle.
(c) Properties of special
quadrilaterals.
Grade 11
Grade 12
(a) Investigate and prove theorems of
the geometry of circles assuming
results from earlier grades, together
with one other result concerning
tangents and radii of circles.
(a) Revise earlier (Grade 9) work on the necessary
and sufficient conditions for polygons to be similar.
(b) Solve circle geometry problems,
providing reasons for statements
when required.
(c) Prove riders.
(b) Prove (accepting results established in earlier
grades):
• that a line drawn parallel to one side of a triangle
divides the other two sides proportionally (and the
Mid-point Theorem as a special case of this
theorem);
• that equiangular triangles are similar;
• that triangles with sides in proportion are similar;
• the Pythagorean Theorem by similar triangles;
• riders.
REVISION FROM EARLIER GRADES
SIMILARITY
AAA
or
∠∠∠
SSS
CONGRUENCY
SSS
AAS
SAS
(included angle)
RHS
PROPERTIES OF SPECIAL QUADRILATERALS
PARALLELOGRAM
• Both pairs of opposite sides are parallel
• Both pairs of opposite side are equal
• Both pairs of opposite angles are equal
• Diagonals bisect each other
RECTANGLE
All properties of parallelogram PLUS:
• Both diagonals are equal in length
• All interior angles are equal to 90°
RHOMBUS
All properties of parallelogram PLUS:
• All sides are equal
• Diagonals bisect each other perpendicularly
• Diagonals bisect interior angles
SQUARE
All properties of a rhombus PLUS:
• All interior angles are 90°
• Diagonals are equal in length
KITE
•
•
•
•
•
Two pairs of adjacent sides are equal
Diagonal between equal sides bisects other
diagonal
One pair of opposite angles are equal
(unequal sides)
Diagonal between equal sides bisects
interior angles (is axis of symmetry)
Diagonals intersect perpendicularly
TRAPEZIUM
• One pair of opposite sides are parallel
HOW TO PROVE THAT A QUADRILATERAL IS A PARALLELOGRAM
Prove any ONE of the following:
• Prove that both pairs of opposite sides are parallel
• Prove that both pairs of opposite sides are equal
• Prove that both pairs of opposite angles are equal
• Prove that the diagonals bisect each other
• Prove that ONE pair of sides are equal and parallel
HOW TO PROVE THAT A PARALLLELOGRAM IS A RHOMBUS
Prove ONE of the following:
• Prove that the diagonals bisect each other perpendicularly
• Prove that any two adjacent sides are equal in length
TRIANGLES BETWEEN PARALLEL LINES
The AREA of two triangles on the SAME (OR EQUAL) BASE between two parallel lines, are EQUAL.
Area of ∆ = Area of ∆
MIDPOINT THEOREM
The line segment joining the midpoints of two sides of a triangle, is parallel to the third side
of the triangle and half the length of that side.
( Midpt Theorem )
If AD = DB and AE = EC, then DE ǁ BC and DE = BC
CONVERSE OF MIDPOINT THEOREM
If a line is drawn from the midpoint of one side of a triangle parallel to another side, that line will bisect
the third side and will be half the length of the side it is parallel to.
( line through midpoint ⃦ to 2nd side )
If AD = DB and DE ǁ BC, then AE = EC and DE =BC.
GRADE 11 GEOMETRY
Note: THEOREMS OF WHICH PROOFS ARE EXAMINABLE ARE INDICATED WITH
Theorem 1
Converse of Theorem 1
If AC = CB in circle O, then OC AB.
(line from centre to midpt of chord)
If OC
chord AB , then AC = BC .
(line from centre
to chord)
Theorem 2
The angle at the centre of a circle subtended by an arc/a chord is double the angle at the circumference
B = 2 ACB
subtended by the same arc/chord.
AO
( ∠ at centre = 2 ×∠ at circumference )
Theorem 3
The angle on the circumference subtended by
the diameter, is a right angle.
The angle in a semi-circle is 90°.
(∠s in semi circle OR
diameter subtends right angle)
Converse of Theorem 3
If = 90°, then AB is the diameter
of the circle.
(chord subtends 90° OR
converse ∠s in semi circle)
Theorem 4
The angles on the circumference of a
circle subtended by the same arc or
chord, are equal.
Converse of Theorem 4
If a line segment subtends equal angles
at two other points, then these four points
lie on the circumference of a circle.
(∠s in the same seg)
(line subtends equal ∠s OR
converse ∠s in the same seg)
Corollary of Theorem 4
Equal chords subtend equal
angles at the circumference
of the circle.
(equal chords; equal ∠s)
∠s)
Equal chords subtend equal angles
at the centre of the circle.
Equal chords of equal circles
subtend equal angles at the
circumference.
(equal chords; equal ∠s)
(equal circles; equal chords; equal
Theorem 5
Converse of Theorem 5
The opposite angles of a cyclic quadrilateral
are supplementary.
= 180°
180°
(opp ∠s of cyclic quad )
If the opposite angles of a quadrilateral
are supplementary, then it is a cyclic
quadrilateral.
( opp ∠s quad sup OR
converse opp ∠s of cyclic quad )
Theorem 6
The exterior angle of a cyclic quadrilateral
is equal to the opposite interior angle.
(ext ∠ of cyclic quad )
Theorem 7
The tangent to a circle is perpendicular to the
radius at the point of tangency.
( tan ⊥ radius OR
tan ⊥ diameter )
Converse of Theorem 6
If the exterior angle of a quadrilateral is equal
to the opposite interior angle, then it is a
cyclic quadrilateral.
(ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad)
Converse of Theorem 7
If a line is drawn perpendicularly to the radius
through the point where the radius meets the
circle, then this line is a tangent to the circle.
( line ⊥ radius OR converse tan ⊥ radius OR
converse tan ⊥ diameter )
Theorem 8
If two tangents are drawn from the same point outside a circle, then they are equal in length.
(tans from common pt OR Tans from same pt )
Theorem 9 (Tan chord theorem)
Converse of Theorem 9
The angle between the tangent to a circle
and a chord drawn from the point of tangency,
is equal to the angle in the opposite circle
segment.
If a line is drawn through the endpoint of
a chord to form an angle which is equal to
the angle in the opposite segment, then this
line is a tangent.
( tan chord theorem )
( converse tan chord theorem OR
∠ between line and chord )
Acute angle
Obtuse angle
THREE WAYS TO PROVE THAT A QUADRILATERAL IS A CYCLIC QUADRILATERAL
Prove that :
•
•
•
one pair of opposite angles are supplementary
the exterior angle is equal to the opposite interior angle
two angles subtended by a line segment at two other vertices of the quadrilateral, are equal.
GRADE 12 GEOMETRY
The Concept of Proportionality (Revision)
A
6 cm
B
D
4 cm
9 cm
AB : BC = 6 : 4 = 3 : 2
C
E
and
6 cm
F
DE : EF = 9 : 6 = 3 : 2
Although, AB : BC = DE : EF it does NOT mean that AB = DE, AC = DF or BC = EF.
Theorem 1
Converse of Theorem 1
A line drawn parallel to one side of a triangle
that intersects the other two sides, will divide
the other two sides proportionally.
( line || one side of Δ
OR prop theorem; name || lines )
If a line divides two sides of a triangle proportionally, then the line is parallel to the third
side of the triangle.
( line divides two sides of Δ in prop )
If DE ǁ BC then =
or
!
AD : DB = AE : EC
If
Theorem 2 (Midpoint Theorem)
(Special case of Theorem 1)
The line segment joining the midpoints of two
sides of a triangle, is parallel to the third side
of the triangle and half the length of that side.
( midpt theorem )
If AD = DB and AE = EC, then DE ǁ BC and DE =BC
=
!
then DE ǁ BC.
Converse of Theorem 2
If a line is drawn from the midpoint of one
side of a triangle parallel to another side, that
line will bisect the third side and will be half
the length of the side it is parallel to.
( line through midpt || to 2nd side )
If AD = DB and DE ǁ BC, then AE = EC and DE =BC.
Theorem 3
Converse of Theorem 3
The corresponding sides of two equiangular
triangles are proportional and consequently
the triangles are similar.
If the sides of two triangles are proportional,
then the triangles are equiangular and
consequently the triangles are similar.
( ||| Δs OR equiangular Δs )
( Sides of Δ in prop )
If ∆ ||| ∆"# then
!
$
!
$
If
!
$
!
$
then ∆ |||∆"#
Theorem 4
The perpendicular drawn from the vertex of the right angle of a right-angled triangle, divides the triangle in
two triangles which are similar to each other and similar to the original triangle.
Corollaries of Theorem 4
∆|||∆
∴
!
!
∴ ' '. '
∆|||∆
∴
!
!
!
!
∆|||∆
∴ ) ). )
Theorem 5 (The Theorem of Pythagoras)
From the corollaries it can be proven that:
∴
!
!
∴ * *. *
TIPS TO SOLVING GEOMETRY RIDERS
• READ-READ-READ the information next to the diagram thoroughly
• TRANSFER all given information to the DIAGRAM
• Look for KEYWORDS, e.g.
TANGENT: What do the theorems say about tangents?
CYCLIC QUADRILATERAL: What are the properties of a cyclic quad?
• NEVER ASSUME something!
- Don’t assume that a certain line is the DIAMETER of a circle unless it is clearly
state or unless you can prove it
- Don’t assume that a point is the CENTRE of a circle unless it is clearly stated
(“circle M” means “the circle with midpoint M”)
• Set yourself “SECONDARY” GOALS, e.g.
- To prove that = (primary goal), first prove that
= (secondary goal) and vice versa
-
To prove that line AC is a tangent (primary goal), first prove that the line is
perpendicular to radius OB (secondary goal)
AC is tangent
-
To prove that BC is the diameter of the circle (primary goal), first prove that =
90° (secondary goal)
BC is the diameter of the circle
• For questions like: Prove that . Start with ONE PART.
Move to the OTHER PART step-by-step stating reasons.
Remember it has to be clear and logical to the reader!
E.g. = ; = ; = ; ∴ = GRADE 11 GEOMETRY SAMPLE QUESTIONS
Question 1
AB and CD are two chords of the circle with centre O.
+"⏊ , AF = FB, OE = 4 cm, OF = 3 cm and AB = 8 cm.
Calculate the length of CD.
[8]
Question 2
O is the centre of the circle. STU is a tangent at T.
BC = CT
- 105° and -/ = 40°
Calculate, giving reasons, the size of:
2.1
2.2
2.3
2.4
(2)
(2)
(3)
(6)
[13]
Question 3
3.1
3.2
Write down with reasons four other angles
which are equal to 1.
Prove that ∆ABC||| ∆EDC.
3.3
Prove that = !. !
!
(8)
(4)
(2)
[14]
Question 4
O is the centre of the circle. BC = CD
Express the following in terms of 1:
4.1
4.2
4.3
(2)
(3)
(4)
[9]
Question 5
LOM is the diameter of circle LMT. The centre of
the circle is O. TN is a tangent at T.
23⏊34
Prove that:
5.1
MNPT is a cyclic quadrilateral.
5.2
NP = NT
(3)
(6)
[9]
Question 6
PA and PC are tangents to the circle at C and A.
AD ǁ PC and PD intersects the circle at B.
Prove that:
6.1
bisects 4
6.2
5
6.3
4 (6)
(6)
(4)
[16]
Question 7
TA is a tangent to the circle. M is the centre of chord PT.
-⏊4. O is the centre of the circle.
Prove that:
7.1
MTAR is a cyclic quadrilateral.
(3)
7.2
PR = RT
(4)
7.3
TR bisects PTA
(4)
(4)
7.4
- +
[15]
GRADE 12 GEOMETRY SAMPLE QUESTIONS
Example
8
Given: : 2: 3 and " ".
5
Instruction: Determine the ratio of 4: 4.
Solution:
In ∆":
9
:
:
=
∴ '; = <"
8
But it was given that '; = "
8
5
:
∴ " = <"
5
!
9
:
8
:
5
>
= ÷ =
In ∆< :
!?
?
=
4: 4 = 15: 8
!
9
=
:
>
Question 1
" = 22@A, 33@A, 15@ACDE 1.
Calculate the value of 1.
[4]
Question 2
5
#||", = > CDE": " 4: 3
Determine the ratio F: F.
[8]
Question 3
G
GH
5 , 4: I 1: 2CDE4J||.
3.1
Write down the values of I: I4and I: K. (2)
3.2
Determine J: I
(1)
3.3
Prove that IJ JK.
(6)
[9]
Question 4
Given: 4KL|CDE4I|L
Prove that KI||.
[4]
Question 5
∆4K-is inscribed in a circle. +||KI, 4 = KCDE4 PR is the diameter of the circle.
Prove that:
5.1
5.2
5.3
||KO is the centre of the circle
BORT is a trapezium.
(2)
(2)
(2)
[6]
Question 6
Given: 4: K = 5: 4CDE4: I 5: 2
S is the midpoint of AQ
6.1
Prove that - 2MI
6.2
If I<||KN, determine 4N: N-
(8)
(6)
[14]
Question 7
Rectangle DEFK is drawn inside right-angled ∆ABC.
Prove that:
7.1
. = ". <
7.2
": " = ": #
7.3
<#: " = ": #
7.4
!
=
(4)
(4)
(1)
(3)
[12]
Question 8
ABOC is a kite with = = 90°
8.1
Why is ∆+|||∆+?
8.2
Complete:
8.2.1 + =. . . …
8.2.2 =. . . …
8.2.3 =. . . …
P
QP
=
Q
8.3
Prove that
8.4
Prove that + − + = +. 8.5
If + = = 1, prove that = √2. +
(2)
(3)
(3)
(2)
(2)
[12]
MIXED EXERCISES
1.
In the diagram, TBD is a tangent to circles BAPC and BNKM at B.
AKC is a chord of the larger circle and is also a tangent to the smaller circle at K.
Chords MN and BK intersect at F. PA is produced to D.
BMC, BNA and BFKP are straight lines.
Prove that:
a)
MN ǁ CA
b)
∆<J3 is isosceles
c)
d)
9
9?
=
T
T!
DA is a tangent to the circle passing through
points A, B and K.
2.
In the diagram below, chord BA and tangent TC of circle ABC are produced to meet at R.
BC is produced to P with RC=RP. AP is not a tangent.
Prove that:
a)
ACPR is a cyclic quadrilateral.
b)
∆|||∆I4
c)
I =
d)
I. = I. e)
Hence prove that I = I. I
3.
In the diagram alongside, circles ACBN and AMBD
Intersect at A and B.
CB is a tangent to the larger circle at B.
M is the centre of the smaller circle.
CAD and BND are straight lines.
Let 5 = 1
a)
in terms of 1.
Determine the size of b)
Prove that:
!.G
!
i)
CB ǁ AN
ii)
AB is a tangent to circle ADN.
4.
In the diagram below, O is the centre of circle ABCD.
DC is extended to meet circle BODE at point E.
OE cuts BC at F. Let" = 1.
a)
Determine in terms of 1.
b)
Prove that:
i)
BE=EC
ii)
BE is NOT a tangent
to circle ABCD.
5.
In the diagram alongside, medians AM and CN of ∆ intersect at O.
BO is produced to meet AC at P.
MP and CN intersect in D.
ORǁMP with R on AC.
a)
Calculate, giving reasons, the numerical value of
b)
Use +: J = 2: 3, to calculate the numerical value of
6.
In the diagram, AD is the diameter of circle ABCD.
AD is extended to meet tangent NCP in P.
Straight line NB is extended to Q and intersect AC in M with Q on straight line ADP.
AC ⏊NQ at M.
a)
Prove that NQ ǁ CD.
b)
Prove that ANCQ is a cyclic quadrilateral.
c)
i)
Prove that ∆4|||∆4.
ii)
Hence, complete: 4 = ⋯
d)
Prove that = . 3
e)
If it is further given that PC=MC, prove that
1−
TP
! P
?.?
= !.U
U
U!
.
G?
?!
.
SOLUTIONS TO MIXED EXERCISE
1. a)
b)
c)
= J
= = ∴J
tan chord
∴ J3||
corr ∠s =
= J
alt ∠s
<
< = 3
tan chord
∴ ∆<J3 is isosceles
8 = 3
<
alt ∠s
3 = 5
∠s in same segment
5 = 5
∠s in same segment
8 = 5
∴<
∴ 3<||4
alt ∠s=
∴
U
9
=
U
9?
U
T
But
9
line || to one side of ∆
= T! line || to one side of ∆
U
T
∴ 9? =
T!
d)
5 = 5
∠s in same segment
5 = equal chords subt equal ∠s
∴ 5 = ∴ is a tangent to the circle through A, B and K
2. a)
5 = 4 I
∠s opp equal sides
5 = ext ∠ of ∆
= tan chord
∴ 5 = ∴ = 4I both = 5
b)
c)
ACPR is a cyclic quadrilateral (ext ∠of quad)
In ∆ and ∆I4:
4 = ∠s in same segment
= proven in 2 a
∴ = 4
= I 4
ext ∠of cyclic quad
5
3rd ∠ of∆
∴ ∆|||∆I4∠∠∠
G?
!
G
! I4 !.G
but I4 = I
!
!.G
∴ I =
d)
from 2 b
!
In ∆I and ∆I:
= tan chord
I is common
I = I 3rd angle
∴ ∆I|||∆I∠∠∠
e)
!
!
G!
G
∆W|||
I. = I. !
G?
!
G!
=
=
!
from 2. b)
G
!
G
!.G
=
RC=RP
G!
From 2.d) =
∴
!.G
G!
G!.!
=
G
G!.!
G
∴ I = I. I
3. a)
b. i)
b. ii)
4. a)
b. i)
b. ii)
= 5 = 1
= 180° − 21
J
= 21
∴
T
X
∠s opp equal sides
sum ∠s of ∆
∠ at centre =2x∠circ
90° − 1
180° − (90° − 1 21Y
90° − 1
90° − 1
3
∴ 3
sum ∠s of ∆
ext ∠of cyclic quad
∴ ||3
= 21
= corr ∠s
tan chord
alt ∠s
∴AB is a tangent
∠betw line&chord
5 " 1
∠s in same segment
1
5 ∠s opp = sides
+ 180° − 21 sum ∠s of ∆
= 90° − 1
∠ at centre =2x∠atcircumference
ext ∠of cyclic quad
90° − 1
# 180° − (1 90° − 1Y sum ∠s of ∆
= 90°
In ∆"# and ∆"#:
# = # = 90°
∠sonstr line
BF = FC
FE is common
∆"# ≡ ∆"#
s∠s
BE = EC
∆W ≡
= 90° − 1
sum ∠s of ∆
∴ = ∴ BE is not a tangent ` ≠ b
5. a)
P is midpoint of AC
AB||PM
In ∆3:
medians concur
midpt theorem
U
line || one side of ∆
U!
=
b)
T
?!
=
=
=
=
c) i)
c) ii)
=
?
!
In ∆J4:
Q
b)
=
!
T
QT
G?
6. a)
T
=
QT
QT
G?
BP is a median
?
QT
line || one side of ∆
T
QT
=
5QT
5
= 90°
= 90°
J
∠ in semi ⊙
AM⏊NM
corr ∠s=
|| lines, corr ∠s
tan chord
∴ 3K||
= 3
= =3
∴ ANCQ is a cyclic quad
∠s subt by same line segm
In ∆4 and ∆4:
= tan chord
4 is common
= 4
3rd ∠
∴ ∆4 ||| ∆4 ∠∠∠
4 = 4. 4
In ∆3 and ∆:
= 3
∠s in same segm
= ∠s in same segm
8 = tan chord
=
∠s in same segm
= 3rd ∠
∴ ∆3 ≡ ∆ ∠∠∠
∴ U U
. 3
eY
1−
!.U
d)
!
!
TP
! P
=
! P eTP
! P
T! P
! P
?! P
! P
?.?
Pyth.