Download (1) , (2) and

The Mathematics Department
Stage : 2nd Prep
Date : / /
Sheet (1)
The Medians
1st Term
Remember:
● The median of a triangle is a line segment joining any vertex of the triangle and
the mid point of the opposite side.
● The medians of a triangle are concurrent.
● The point of concurrence of the medians of the triangle divides each median in
the ratio 1 : 2 from the base.
● The line segment joining the midpoints of two sides of a triangle is parallel
to the third side and its length equals half the length of this side.
● The ray drawn from the midpoint of a side of a triangle parallel; to another
side bisects the third side.
[1] Complete:
1] The median of a triangle is …a line segment…joining the vertex of the triangle
and …the midpoint of the opposite side…
2] The intersection point of the medians of a triangle divides each in the ratio
…1:2..from the base, and in the ratio …2:1..from the vertex.
3] In the ∆ ABC, if D is a midpoint of BC then AD is called…the median…
4] The medians of the triangle intersect at…the point of conccurence…..
5] The number of medians of the right angled triangle is…three…
6] The line segment joining the two midpoints of two sides of a triangle is
…parallel…. and its length equals …half the length of the third side…
[2] Complete the missing:
X
Z
YE =…5…cm
F
AM =…6….cm.
DM =…2….cm.
EM =…1/3….AE.
CM =…2/3…CD.
M
4cm
G
If AE = 9 cm , CD = 6 cm
Then:
XM =…8…cm
Y
A
D
M
C
B
E
E
10cm
If BC = 12 cm , BE = 9 cm
MC = 8 cm. then :
If
A
ZL =12cm , YN=15 cm
ZY= 24cm , then:
N
DE =…6…cm
EM =…3….cm
DM =…4….cm
E
M
M
ML=…4…cm
M
Z
MY= …10...cm
Perimeter of ∆ MLN = …21…cm
Perimeter of ∆ MYZ = …42…cm
D
M
C
B
1
X
L
Y
[3] In the opposite figure:
If
D, E are midpoints of AB , AC , BE
A
CD = {M}
and if DE = 4 cm , DM= 3cm , BE= 6cm,
find the perimeter of the ∆ BMC
Given: D & E are the midpoint of AB & AC
BE
E
CD = {M}
DE = 4 cm , DM = 3 cm , BE = 6 cm
R.T.P: The perimeter of the ∆ BMC
Proof: BE
4cm
4cm
3cm D
3cm
M
C
B
CD = {M}
D & E are the midpoint of AB & AC
M is the point of intersection of the medians
BC = 2 DE = 2 × 4 = 8 cm
CM = 2 DM = 2 × 3 = 6 cm
BM =
2
2
BE =
× 6 = 4 cm
3
3
Perimeter of ∆ BMC = BC + CM + BM = 8 + 6 + 4 = 18 cm
[4] Complete:
a) The length of the median drawn from the vertex of the right angled triangle
equal to…half the length of the hypotenuse..
b) The length of the side opposite to the angle 30° in the right angled triangle
equal to… half the length of the hypotenuse …..
c) The length of the hypotenuse in the right angled triangle equal…twice….the side
opposite to the angle 30°.
d) In any right – angled triangle, if the length of hypotenuse equals 8cm then the
side apposite the angle of measure 30 equals…4...cm.
e) In a right – angled triangle, the measure of two angles are 60º and 30º, then
the length of the hypotenuse …twice… the length of the side opposite to the
angle of measure 30º.
A
f) In the opposite figure
X, Y mid- points of AC and BC respectively
mC =30 , mB = 90
AC = 6cm, YM = 1.2 cm
Then AB = …3…cm
AM = …2.4….cm
BM = …2…cm
2
6cm
/
X
/
M
1.2
C
30°
//
Y
//
■ B
[5] In the opposite figure:
D
ABCD is a quadrilateral mB =90 , E is a midpoint of AD
F is a midpoint of CD , mACB =30,
EF = 4cm
Find with proof the length of AB
In ∆ ACD:
E & F are the midpoints of DA & DC
EF = ½ AC
AC = 2 EF = 2 × 4 = 8 cm
In the right angled triangle ABC:
m (ACB) = 30°
E
4cm
4cm
A
F
30°
C
B
AB = ½ AC = ½ × 8 = 4 cm
[6] In the opposite figure:
D
mABC = mADC =90,
mACB = 30, E is a midpoint of AC
Prove that: AB = DE
A
In ∆ ABC: mB = 90° & mACB = 30°
AB = ½ AC
(1)
In ∆ ADC: mADC = 90° & DE is a median
DE = ½ AC
(2)
From (1) & (2)
AB = DE
E
30°
C
B
[7] In the opposite figure:
ABCD is a quadrilateral in which
M is the midpoint of BD ,
E is the midpoint of BC ,
EF // BD where F DC .
If AM = EF, prove that: mBAD = 90º
A
D
/
<
M
F
In ∆ BCD:
E is a midpoint of BD & EF // BD
EF = ½ BD
AM = EF
EF = ½ BD
AM is a median
m BAD = 90°
/
B
< 
\\
C
3

\\
E
[8] In the opposite figure:
F
mYZE = 90, m E = 30,
YE = 10 cm, mXYF = 90,
Z is the midpoint of XF ,
Find with proof the length of XF
Z
X
In ∆ EZY:
mEZY = 90° & mE = 30°
ZY = ½ EY
= ½ × 10 = 5 cm
Y
10cm
30°
E
In ∆ XYF:
mXYF = 90° & YZ is a median
YZ = ½ XF
XF = 2 × 5 = 10 cm
4
The Mathematics Department
Stage : 2nd Prep
Date : / /
1st Term
Sheet (3)
Evaluate Yourself
15
[1] Choose the correct answer:
1] The number of medians in the triangle equals ………
i) 1
ii) 2
iii) 3
5
iv) zero
2] The point of concurrence of the medians of the triangle divides each of them in
the ratio ……. from the vertex.
i) 2:1
ii) 1:2
iii) 2:3
iv) 1:3
3] If ∆ XYZ is right-angled at X, M is the midpoint of YZ , then XM = …
1
1
i) XZ
ii) YZ
iii) XZ
iv) YZ
2
2
4] In the right-angled triangle, the length of the hypotenuse equals …… the length
of the side opposite to the angle whose measure is 30º.
i) twice
ii) half
iii) three times
iv) quarter
5] In the opposite figure:
If AD is a median in ∆ ABC,
then m(BAC) = 90º if ……
1
i) AD = BC
ii) AD = BC
2
A
C
iii) AD = 2BC
[2] In the opposite figure:
mABC = 90°, mC = 30°,
X, Y, Z are midpoints of AB , BC , XY by order,
AC = 8 cm, Find with proof:
The length of AB , XY ,BZ
In ∆ ABC:
●mABC = 90° & mC = 30°
AB = ½ AC
= ½ × 8 = 4 cm
● X & Y are the midpoints of AB & BC
XY = ½ AC
= ½ × 8 = 4 cm
In ∆ XZY:
● m XBZ = 90° & BZ is a median
BZ = ½ XY
= ½ × 4 = 2 cm
5
iv) AD =
1
BC
4
B
D
A
‫ס‬
X
‫ס‬
B∎
5
●
Z
/
●
Y
/
C
[3] In the opposite figure:
ABC is a triangle, where mABC = 90°
mC= 30° , BE ^ AC
A
E
BD is a median in the ∆ ABC , AE = 4cm ,
Calculate with proof the perimeter of the ∆ ABD
In ∆ ABC:
m (ABC) = 90° & m (C) = 30°
AB = ½ AC
(1)
BD is a median
BD = ½ AC
C
D
30°
∎B
(2)
From (1) & (2)
AB = BD = ½ AC = AD
In ∆ ABE & ∆ DBE:
ìï 1)BA = BD
ïï
ïí 2)m(ÐAEB) = m(ÐDEB) = 90° BE ^ AC
ïï
ïï 2)BEis a common side
î
∆ ABE ≡ ∆ DBE
AE = ED = 4 cm
AD = 2 AE = 2 × 4 = 8 cm
AB = BD = AD = 8 cm
Perimeter of ∆ ABD = 8 + 8 + 8 = 24 cm
6
5
The Mathematics Department
Stage : 2nd Prep
Date : / /
1st Term
Sheet (4)
The Isosceles Triangle
Remember:
● The isosceles triangle is a triangle in which two sides are equal in length, these
two sides are called the sides (legs) of the isosceles triangle and the third
side is called the base of the triangle, the angle opposite to the base is called
the angle of vertex, and the two angles opposite to the two congruent sides
are called the base angles
● The equilateral triangle is a triangle in which the three sides are equal in
length.
● The base angles of the isosceles triangle are congruent.
● In the equilateral triangle, the measure of each angle is 60º.
● The measure of an exterior angle of a triangle equals the sum of the
measures of its non-adjacent interior angles.
[1] Complete:
1] ∆ ABC, AB = AC, m (A) = 120º, so m (C) = …30…º
2] ∆ XYZ is isosceles, if m (X) = 100º, so m (Y) = …80….º
3] ∆ ABC is isosceles right-angled at B, the measure of its base angle equals …45º
4] The measure of the base angle of an isosceles triangle is 50º, so the measure of
its vertex angle = …80..º
5] If the three angles in any triangle are congruent, then the triangle is
equilateral
6] In ∆ XYZ, m(X) = 48º and m (Y) = 84º, then the type of the triangle
according to its lengths is …isosceles…
7] DEF is a triangle in which DE = DF, then m (E) = m (…F….)
8] In the isosceles triangle, if the measure of the vertex angle is 40°, then the
measure of one of its base angle equals …70°…..
9] If two angles in a triangle are congruent, then the two opposite sides to these
angles are …equal in length…. and the triangle is …isosceles…..
10] If the measure of one angle in a right-angled triangle is 45°, then the triangle
is … isosceles …
11] If the measure of an angle in an isosceles triangle is 60°, then the triangle is
…equilateral triangle….
7
12] In ∆ ABC, AB = AC and m (A) = 60°, if its perimeter equals 15 cm, then BC
= …5… cm
13] Using the given data for each figure, complete to fill the spaces under each
figure:
1]
3]
2]
\\
//
x
\\
x = 20°
x = 56°
5]
x
//
2x
62°
x = 50°
4]
\\
//
\\
50°
5x
x
6]
42°
//
y
\\
y
63°
x = 54° & y =
117°
//
\\
x
y
x = 69° & y = 111°
//
x
||
x = 60° & y = 120°
A
\
[2] In the given figure:
AC = CB = CD
, mB = 36 , find mD
36°
In ∆ ACB:
B
\
C
\
D
AC = CB Þ mCAB = mCBA = 36°
, mACD = 36° + 36° = 72°
(exterior angle)
In ∆ ACD: CA = CD Þ
mCAD = mCDA = (180° -72°)  2 = 54°
[3] In the given figure:
AB = AC = CD, m ( BAC) = 30
, m ( CAD) = 64.
Find: m ( BCD)
A
30° 64°
D
\\
//
\\
In ∆ ABC: AB = AC, m BAC = 30
mACB = (180 - 30)  2 = 75°
B
(1)
In ∆ CAD: CA = CD, m  CAD = 64
mCAD = mCDA = 64°
mACD = 180° - (64° + 64°) = 52°
mBCD = mBAC + mACD = 75°+52°=127°
8
C
[4] In the given figure:
AB = AC, DB = DC
m A = 80
mDBC = 30
Find: m ABC , mABD , m  BDC
A
80°
\\
//
D
/
\
In ∆ AB = AC: AB = AC Þ
30°
B
m(ABC) = m(ACB) = (180° - 80°)  2 = 50°
C
(1)
m(ABD) = 50° - 30° = 20°
In ∆ DBC: DB = DC
m(DBC) = m(DCB) = 30
m(BDC) = 180° - (30° + 30°) = 120°
[5] In the given figure:
AB = AC, DE // AB
, m (A) = 34 Find: m (ACB)
and m (CDE)
E
<
D
C
In ∆ ABC:
\\
AB = AC and m (A) = 34
34°
A
m(ACB) = m(B) = (180° - 34°)  2 = 73°
\\ <
B
(1)
DE // BA Þ
m(D) + m(B) = 180°
(interior angles)
m(CDE) = 180° - 73° = 107°
[6] In the given figure:
E  AB , AD = BC, m (A) = m (B)
& m (DEC) = 40°.
Find with proof: m (EDC)
C
D
=
=
In ∆ ADE , ∆ BCE:
ìï 1)AD = BC
ïï
ïí 2)AE = BE
ïï
ïïî 3)m(ÐA) = m(ÐB)
∆ ADE ≡ ∆ BCE
DE = CE Þ m(EDC) = m(ECD)
In ∆ EDC:
m (DEC) = 40
m (EDC) = (180 – 40)  2 = 70°
40°
B
9
●
|
E
|
● A
[7] In the given figure:
AB = AC, and DB = DE
Prove that:
AC // ED
A
10
E
/
/
B
D
//
\\
In ∆ ABC: AB = AC
m(ABC) = m(C)
(1)
In ∆ DBC:
DB = DE Þ m(DBE) = m(E)
(2)
m(ABC) = m(DBE)
(3)
From (1) , (2) and (3)
m(C) = m(E)
They are alternate angles Þ AC // ED
C
The Mathematics Department
Stage : 2nd Prep
Date : / /
1st Term
Sheet (5)
Corollaries Related to the
isosceles triangle
Remember:
● The median of the isosceles triangle, which is drawn from the vertex angle,
bisects the angle and is perpendicular to the base.
● The bisector of the vertex angle of an isosceles triangle bisects the base and
is perpendicular to it.
●The straight line drawn from the vertex angle of an isosceles triangle and
perpendicular to the base bisects each of the base and the vertex angle.
● The axis of symmetry of a line segment is its perpendicular bisector.
● Every point on the axis of symmetry of a line segment is equidistant from its
end-points.
● If a point is equidistant from the end point of a line segment, then this point
lies on the axis of symmetry of this line segment.

The axis of symmetry of the isosceles triangle is the straight line drawn from
the vertex angle perpendicular to the base.
[1] Choose the correct answer:
1] The isosceles triangle has …. axis of symmetry.
i) one
ii) two
iii) three
iv) four
2] AB = AC and m (B) = 40º in the triangle ABC, then the exterior angle at its
vertex is ….. angle.
i) acute
ii) obtuse
iii) right
iv) straight
3] The measure of the exterior angle of the equilateral triangle = ……..º
i) 30
ii) 150
iii) 90
iv) 120
4] If ∆ ABC has only one axis of symmetry and mABC = 120º, then mA = …º
i) 60
ii) 120
iii) 30
iv) 40
5] The symmetric axis of the line segment is any line ……….
i) Perpendicular to it.
ii) bisects and perpendicular to it.
iii) cut the line segment.
iv) perpendicular to it at one of its two end points
[2] Complete each of the following:
1] The number of symmetry axes of the equilateral triangle is …3..
2] The number of symmetry axes of the isosceles triangle is …1..
11
3] The number of symmetry axes of the scalene triangle is …zero..
4] The median of the isosceles triangle drawn from the vertex angle bisects
……bisects this angle…. and is …perpendicular to its base….
5] The symmetry axis of the line segment is the straight line that …is
perpendicular to it from its midpoint….
6] Any point belongs to the axis of a line segment is …equidistant.. from its
terminals
7] If the point C belongs to the symmetry axis of the line segment AB , then
…CA.. = …CB….
[3] In the given figure:
m (ABD) = 110 and
m (BAE) = 140
Prove that:  ABC is an isosceles triangle
C
B
110°
m (ABC) + m (ABD) = 180°
m (ABC) = 180° - 110 = 70°
m (BAC) + m (BAE) = 180
m (BAC) = 180° - 140° = 40°
In ∆ ABC: m (C) = 180° - (70 + 40) = 70°
m (C) = m (ABC) = 70°
AB = AC Þ  ABC is an isosceles triangle
A
140°
D
E
A
[4] In the given figure:
DB = DC and
m ( ABD) = m (ACD)
Prove that:  ABC is an isosceles.
B
C
//
\\
In ∆ DBC: DB = DC
m (DBC) = m (DCB)
(1)
m ( ABD) = m (ACD)
(2)
(2) – (1) Þ m (ABC) = m (ACB)
AB = AC Þ
 ABC is an isosceles.
D
[5] In the given figure:
AB = AC and
m ( ABD) = m (  ACD)
Prove that: DB = DC
A
\\
In ∆ABC: AB = AC
m(ABC) = m(ACB)
(1)
m ( ABD) = m (  ACD)
(2)
(1) – (2) Þ m(DBC) = m(DCB)
DB = DC
●
B
12
D
//
●
C
[6] In the given figure:
XY // AC , m (ABX) = 62
and m (C) = 56
Prove that: AC = BC
A
X
62°
XY // AC
m(A) = m (ABX) = 62 (alternate angles)
In ∆ ABC:
m (ABC) = 180° - (62° + 56) = 62°
m(A) = m(ABC)
AC = BC
[7] In the given figure:
ABCD is a quadrilateral
AB = AD, AD // BC
,m (ABD) = 36
m (C) = 72
Prove that: BC = BD
B
56°
C
Y
D
A
< \\
//
36°
C
72°
<
B
In ∆ ABD:
AB = AD
m (ABD) = m(ADB) = 36
AD // BC
m(ADB) = m(DBC) = 36° (alternate angles)
In ∆ DBC:
m(BDC) = 180° - (72° + 36°) = 72°
m(BDC) = m(C)
BC = BD
[8] In the given figure:
ABC is a right-angled triangle at B in which m
(A) = 45°, BD bisects B and AD = 20 cm.
i) Find the length of AC
ii) Prove that the triangle DBC is an
isosceles triangle
In  ABD
m (A) = 45°, m (B) = 90° and BD bisects B
m (ABD) = 90° ÷ 2 = 45°
m (ADB) = 180° - (45° + 45°) = 90°
BD  AC
In  ABC
m (C) = 180° - (45° + 90°) = 45°
 ABC is isosceles
BD  AC
→
D is a midpoint
AD = 20 cm
→
AC = 40 cm
13
A
D
C
●
●
B
[9] In the given figure:
ABCD is a trapezium in which AD // BC , and
AE bisects A. BD bisects B
Prove that:
i) AB = AD
ii) AE ^ BD
iii) BE = ED
D
In  ABD
m (ADB) = m (ABD)
AE bisects A
 ABD is isosceles
AE  DB
E is midpoint
C
(Alternate angles)
AB = AD
→
BE = ED
14
★
★
E
AD // BC
m (ADB) = m (DBC)
m (ABD) = m (DBC)
m (ADB) = m (ABD)
A
<
<
●
●
B
The Mathematics Department
Stage : 2nd Prep
Date : / /
1st Term
Sheet (6)
Evaluate Yourself
15
[1] Choose the correct answer:
[a] In the equilateral triangle , the measure of each angle equals……..…
1) 30
2) 60
3) 45
4) 90
5
[b] If the measure of one angle of a right angled triangle is 45, then the triangle
is……
1) equilateral
2) isosceles
3) scalene
4)obtuse angled
[c] If the triangle ABC is a right – angled  at B. m(A) =
1) 30
2) 60
3) 45
1
m(C), then m(A) =..
2
4) 90
[d] The bisector of the angle at the vertex of an isosceles triangle ……
1) intersects the base
2) bisects the base
3) only perpendicular on it
4) bisects the base and  to it
[e] The lengths 8 , 17 , ……..are the lengths of an isosceles triangle.
1) 8
2) 9
3) 17
4) 2
[2] Complete:
[a] If the measure of the angle at the vertex of an isosceles  is 80,
then the measure of each angle of the base is …50°…
5
[b] The base angles of isosceles triangle are …congruent….
[c] If the angles of a triangle are congruent, then the triangle has …3…. axes of
symmetry
A
[d] Complete using the figure:
AB = AC
Then m (ABC) = …72..
m ( ABE) = …108…
36°
\\
E
[3] In the given figure:
CD = {E}.
AC = AD, BC = BD and AB
Prove that:
i) m(CAB) = m(DAB)
ii) AE ^ CD
In ∆ ADB, ∆ ACB:
D
ìï 1- AD = AC
given
ïï
ïí 2- BD = BC
given
ïï
ïïî 3- AB isa common side
∆ ADB ≡ ∆ ACB
 m(1) = m(2)
In ∆ADC: AD = AC, AE bisects A
 AE  DC
15
//
C
B
A
\\
B
\
//
5
/
E
C
The Mathematics Department
Stage : 2nd Prep
Date : / /
1st Term
Sheet (7)
Inequalities in Triangles
Remember:
● In a triangle, if two sides have unequal lengths, then the longer side is opposite
to the angle of greater measure.
● In a triangle, if two angles are unequal in measure, then the greater angle in
measure is opposite to the greater side in length.
●The hypotenuse is the longest side in the right-angled triangle.
● The shortest possible distance, between a given straight line and a given point
outside it, is the length of the perpendicular from this point to this straight line.
● The sum of the lengths of any two sides of a triangle is greater than the
length of the third side.
[1] Complete the following:
1) In a triangle, if two sides have unequal lengths, then the longer side is
…opposite to the angle of greatest measure…
2) In a triangle, if two angles are unequal in measure, then the greater angle in
measure is opposite to …a greater side in length…..
3) The largest side in length in the right-angled triangle is ..the hypotenuse…
4) In any triangle, the sum of the lengths of any two sides ……………………….
…is greater than the length of the third side.
5) If the lengths of two sides in an isosceles triangle are 8 cm and 4 cm, then the
third side is of length …8 cm
6) In ∆ ABC, if AB > AC > BC, then m (..A..) < m(..B..) < m(..C..)
7) The shortest distance between a given straight line and a given point outside it
is … the length of the perpendicular from this point to the straight line.
8) Let ∆ ABC such that: m ( ÐA) = 50 and m (B) = 70º, then AB > …BC…
9) Let ∆ ABC such that Ð C is an obtuse angle, then the greatest side in length of
this triangle is … AB ….
10) In ∆ DHO, if DH < HO, then …mO < mD…
11] In ∆ ABC, if AB = 6 cm, BC = 7 cm, then C ϵ ]…1., …13.[
12] In ∆ AB = 2.5 cm, BC = 7.5 cm, then AC ∈ ] …5.. , …10.. [
16
[2] In the opposite figure:
ABC is triangle in which AB = 5 cm
BC = 4.8cn, AC = 4.2cm
Complete:
m (…B..) < m (…A..)< m (…C…)
A
4.2cm
5cm
C
B
4.8cm
[3] Which of the following sets of numbers can be side lengths of triangle
1) 4 , 6 , 11
3) 13 , 8 , 6
2) 3 , 5 , 10
4) 7 , 8 , 18
[4] In the opposite figure:
ABC is triangle and B  CD
Where AB = BD
Prove that CD > AC
A
//
C
In ∆ ABC:
CB + BA > AC
AB = BD
CB + BD > AC
CD > AC
D
//
B
A
[5] In the given figure:
AX > AY and XY // BC
Prove that AB > AC
X
<
B
In ∆ AXY: AX > AY Þ m(2) > m(1)
XY // BC , AB is a transversal Þ m(1) = m(B)
XY // BC , AC is a transversal Þ m(2) = m(C)
m(C) > m(B)
AB > AC
Þ
2
1<
17
C
(c0rresp)
(c0rresp)
[6] In the given figure:
M is a point inside  ABC
Prove that:
MA + MB + MC > ½ the perimeter of  ABC
In ∆ ABM: MA + MB > AB (1)
In ∆ BMC: MB + MC > BC
(2)
In ∆ AMC: MA + MC > AC
(3)
(1) + (2) + (3)
2MA + 2MB + 2MC > AB + BC + AC
MA + MB + MC > 1/2 the perimeter of  ABC
Y
A
M
B
C
[7] In ∆ ABC, if m (A) = (3x)°, m (B) = (3x – 5)° and m (C) = (4x – 15)°.
Arrange the side length ascendingly
m (A) + m (B) + m (C) = 180°
3x + 3x – 8 + 4x – 12 = 180°
10x – 20 = 180°
10x = 180 + 20
10x = 200
x = 20
m (A) = 3(20) = 60°
m (B) = 3(20) – 8 = 52°
m (C) = 4(20) – 12 = 68°
m (B) < m (A) < m (C)
→
AC < BC < AB
[8] In ∆ ABC, if m (A) = (2x)°, m (B) = (3x)°, m (C) = (x – 6)°. Arrange the
side length descending
m (A) + m (B) + m (C) = 180°
2x + 3x + x – 6 = 180°
6x – 6 = 180°
6x = 180 + 6
6x = 186
x = 31
m (A) = 2(31) = 62°
m (B) = 3(31) = 93°
m (C) = (31) – 6 = 25°
m (B) > m (A) > m (C)
→
18
AC > BC > AB
The Mathematics Department
Stage: 2nd Prep
Date : / /
1st Term
Sheet (8)
Evaluate Yourself
15
[1] Choose the correct answer:
1) If ∆ ABC is a right-angled triangle at B, then AC ….. BC.
i) >
ii) =
iii) <
iv) ≡
5
2) ∆ ABC in which AB = AC and m(BAC) = 50º, then ……..
i) BC > AB
ii) BC = AB
iii) BC < AB
iv) BC ^ AB
3) Which of the following numbers can be side lengths of a triangle?
i) 3.5, 7.5, 11 ii) 5.6, 4.3, 12
iii) 3.2, 8.4, 10.5
iv) 3, 12, 8
4) If 3 cm and 7 cm are two side lengths of an isosceles triangle, then the third
side length is…………
i) 3 cm
ii) 7 cm
iii) 10 cm
iv) 4 cm
5) ABC is a triangle in which: m(B) = 60º and m(C) = 40º, then the longest
side is ……
i) AB
ii) BC
iii) AC
iv) otherwise
[2] In the opposite figure:
m(BAC) = 75º, m(DAC) = 35º
and AD // BC .
Prove that: BC > AC
AD // BC
m(DAC) = m(C) = 35°(alternate angles)
In ∆ ABC
m(B) = 180° — (75° + 35°) = 70°
m(BAC) > m(B)
 BC > AC
A
<
35°
75°
4
<
C
B
A
/
4
B
\
[3] In the opposite figure:
ABCD is a quadrilateral,
AB = AD, CD = 2 cm and BC = 3 cm.
Prove that: m(ADC) > m(ABC)
In ∆ ABD:
AB = AD
(1)
m(1) = m(2)
In ∆CBD:
CB > CD
 m(3) > m(4)
(2)
By adding (1) and (2)
m(1) + m(3) > m(2) + m(4)
m(ADC) > m(ABC)
D
3cm
D
2cm
C
[4] In ∆ ABC, if AB = 3.5 cm, BC = 9.5 cm, find the interval in which the length of
AC belongs
Let the length of the third side be x
9.5 – 3.5 < x < 9.5 + 3.5
6 < x < 13
∴ x ϵ ]6, 13[
2
19
The Mathematics Department
Stage: 2nd Prep
1st term
Final Revision "Geometry"
Unit One: Medians of a Triangles
& the Isosceles triangle
[1] Choose the correct answer:
1] In the isosceles triangle the number of axes of symmetry is ……….
a) one
b) two
c) three
d) zero
2] If the measure of two angles in a triangle are 70° and 40°, then the triangle is
………..
a) equilateral
b) isosceles
c) scalene
d) right-angled
3] The bisector of the vertex angle in the isosceles triangle ………….
a) only cuts the base
b) bisects the base & is perpendicular to it
c) bisects the base only
d) is perpendicular to the base only
4] In the equilateral triangle the number of axes of symmetry is ……….
a) one
b) two
c) three
d) zero
5] In the isosceles triangle if the measure of one of its base angles is 50°, then the
measure of the vertex angle equals ……….°
a) 50
b) 70
c) 65
d) 80
6] In ∆ ABC: If m (  B) = 45°, m (  C) = 90°, then the number of axes of symmetry
is ……….
a) 1
b) 2
c) 3
d) zero
7] In ∆ ABC : if m (  B) = 90°, m (  A) = 60°, then AC = ……….
a) BC
b) AB
c) ½ AB
d) 2 AB
8] AD is a median in the right-angled triangle ABC at A, M is the intersection point of
the medians. If BC = 12 cm, then AM = ……. cm
a) 6
b) 3
c) 4
d) 8
9] The point of intersection of the medians of a triangle divides each of them in ratio
…….. from the vertex.
a) 1 : 2
b) 2 : 1
c) 3 : 1
d) 3 : 2
10] If ABC is a triangle in which: AB = AC and m (  A) = 60°, then the number of
axes of symmetry in it is ………….
a) 1
b) 2
c) 3
d) zero
11] The axis of symmetry of a line segment is the straight line …………
a) perpendicular to it
b) that bisects it
c) that is parallel to it
d) perpendicular to it from its midpoint
12] The measure of the exterior angle of the equilateral triangle = …….°
a) 180
b) 60
c) 120
d) 90
20
13] The length of the hypotenuse of the right-angled triangle equals …….. the length
of the median from the vertex of the right angle.
1
1
1
a)
b)
c)
d) 2
2
4
3
14] The length of the side opposite to the angle of measure 30° in the right-angled
triangle equals …….. the length of the hypotenuse.
1
1
1
a)
b)
c)
d) 2
2
4
3
15] The number of medians in the triangle equals ………..
a) 1
b) 2
c) 3
d) zero
[2] Complete the following:
1] In ∆ ABC, if m (A) = 70°, m (B) = 30°, then the greatest side of the triangle is
… AB …
2] If M is the point of intersection of the medians in ∆ ABC, AD is a median whose
length is 9 cm, then AM = …6… cm.
3] In the right-angled triangle, the length of the hypotenuse equals … twice… the
length of the side opposite to the angle whose measure is 30°.
4] The two base angles in the isosceles triangle are …congruent….
5] The straight line that is perpendicular to a line segment from its midpoint is called
… its axis of symmetry….
6] In ∆ ABC, if m (  A) = m (  B) = 60°, then the number of symmetry axes to this
triangle = …3…
7] If ∆ XYZ has only one symmetry axis and m (  XYZ) = 120°, then m (  X) = 30..°
8] The point that is equidistant from the two terminals of a line segment belongs to
…… the axis of symmetry…..
9] The point of intersection of the medians of the triangle divides each of them with
the ratio …2.. : …1.. from the vertex
10] In ∆ ABC, if D is the midpoint of BC , then AD is called …the median…
11] The number of medians in the right-angled triangle is …3…
12] The line segment drawn between the two midpoints of two sides in a triangle is
…parallel to the third side…. and its length equals …half its length…
13] XY // ZL and XY = ZL, then the quadrilateral XYZL is called …parallelogram….
21
14] The length of the side opposite the angle whose measure is 30° in the rightangled triangle = …half the length of the hypotenuse….
15] The medians of the triangle intersect at …at one point..
16] In ∆ ABC right-angled triangle at B, if D is the mid-point of AC and AC = 8 cm
then BD = …4 cm..
17] In ∆ ABC, if D is the mid-point of AC and BD = ½ AC, then m (B) = …90°…
[3] Using data given for each the following figures, find the required below each
figure:
A
A
//
10 cm
D
5 cm
//
B■
C
B
AC = …10.. cm
■
■
30°
C
AB = …5.. cm
A
A
//
D
9 cm
//
M
B■
×
14 cm
D
//
30°
//
×
E
B■
C
AC = …18.. cm
BD = …9.. cm
MD = …1/3.. BD & MD = …3…. cm
A
A
E
D
E
\\
\\
●
\\
\\
●
M
B
If BC = 12 cm, BE = 9 cm & MC = 8 cm
DE = …6…. cm
ME = …3…. cm
MD = …4…. Cm
F
●
■
B
D
If AB = 8cm, BC =10cm & AC = 9cm
DE = …4.5.. cm
DF = …4... cm
FE = …5… cm
Perimeter of ∆ DEF = …13.5….. cm
[4] In the following figures, name the equal sides:
A
2y
50°
●
C
C
2y + 3y + 5 + 50 = 180°C
5y + 55 = 180°
5y = 180 – 55 = 125°
y = 125 ÷ 5 = 25
m(A) = 2×25 = 50°
m(B) = 3×25+5 = 80°
m(A) = m(C)
BA = BC
C
BD = …7.. cm
AB = …7.. cm
The perimeter of ∆ ABD = ..21… cm
3y+5°
B
AD // BC
m(C) = m(CAD) = 70°
2x - 5 + x + 25 + 70 = 180°
C
3x + 90 = 180°
3x = 180 – 90 = 90°
x = 90 ÷ 3 = 30°
m(A) = 2×30 - 5 = 55°
m(B) = 30 + 25 = 55°
m(A) = m(B)
AB = AC
22
A
70°
2x-5
x+25°
B
[5] In the opposite figure:
A
∆ ABC in which AB = AC, BX = CY
, m (C) = 50° and m (BAX) = 30°
i) Prove that: ∆ XYA is isosceles triangle
\\
ii) Find m (AYC)
In ∆ ABC: AB = AC
m (B) = m (C) = 50°
50°
m (AXB) = 180° - (50° + 30°) = 100°
C
/
Y
In ∆ ABY , ∆ ACX:
ìï 1. AB = AC
given
ïï
ïí 2. BX = CY
given
ïï
ïïî 3. m (ÐB) = m (ÐC) proved
∆ ABY  ∆ ACX
m(AYB) = m(AXC)
∆ AXY is isosceles triangle
AYC supplements AYX & AXB supplements AXY
m (AYC) = m (AXB) = 100°
[6] In the opposite figure:
ABC is an isosceles triangle in which
AB = AC, BD bisects ABC, CD bisects  ACB.
Prove that: ∆ DBC is isosceles.
In ∆ ABC: AB = AC
m(ABC) = m(ACB)
1/2 m(ABC) = 1/2 m(ACB)
C
m(DBC) = m(DCB)
DB = DC
∆ DBC is isosceles.
//
\
X
B
A
\\
//
D


●
●
[7] In the opposite figure:
AB = BC, AD = 20 cm.
BD  AC .
1) Find the length of AC
2) Prove that: ∆BCD is isosceles
In ∆ ABC: AB = BC and BD ^ AC
D is the midpoint of AC
AD = 20 cm Þ AC = 2 × 20 = 40 cm
In ∆ ABC: m(B) = 90° and BD is a median
BD = 1/2 AC and DC = 1/2 AC
BD = DC
∆BCD is isosceles
B
A
20cm
D
\
[8] In the opposite figure:
∆ ABC in which AB = BC, BC bisects ABC and cuts
AC at D. Draw DE // CB and DE ∩ AB = {E}.
Prove that: BE = ED
BA = BC & BD bisects B
BD bisects AC
D is the mid point of AC & DE // CB
E is the mid point of AB &
DE = ½ BC
BE = ½ AB = ½ BC
(2)
From (1) & (2)
BE = ED
23
30°
B
■
\
C
A
D
C
(1)
>
>
E
●
●
B
The Mathematics Department
Stage : 2nd Prep
1st term
Final Revision "Geometry"
Unit Two: Inequalities in Triangles
[I] Choose the correct answer:
1] The sum of two lengths of two sides in a triangle …….. the length of the third side.
i) >
ii) <
iii) =
v) ≥
2] The set whose elements can be lengths to the sides of a triangle is ……….
i) {3,4,8}
ii) {3,4,7}
iii) {3,4,6}
v) {3,5,8}
3] If ABC is a right angled triangle at B, then AC …… AB.
i) >
ii) <
iii) =
v) ≤
4] In ∆ ABC : if AB = 4 cm, BC = 6 cm, AC = 5 cm, then the greatest angle in
measure is ……
i) A
ii) B
iii) C
v) other wise
5] If the lengths of two sides in an isosceles triangle are 3 cm and 8 cm, then the
length of the third side = ………cm.
i) 3
ii) 8
iii) 5
v) 11
6] In ∆ ABC if m (B) = 100° and m (C) = 40°, then AB ….. BC.
i) >
ii) <
iii) =
v) ≠
7] If X > Y and Z < Y, then ………….
i) X > Z
ii) Y < Z
iii) X < Y
v) Z > X
8] If a triangle has one symmetry axis and the lengths of two sides of it are 4 cm and
8 cm, then its perimeter equals ……….cm.
i) 30
ii) 24
iii) 16
v) 20
9] If the measure of two angles in a triangle are unequal, then the greatest in
measure is opposite to a side ……… in length than the side opposite to the other.
i) smaller
ii) greater
iii) equal
v) smaller than or equal
10] In ∆ ABC : m (C) = 65° and m (B) = 45°, then …………..
i) AC > BC
ii) AC > AB
iii) AB > AC
v) BC < AB
11] If 5 cm, 10 cm are the length of two sides of a triangle, then the length of the
third side belongs to ……..
i) [5, 15[
ii) ]5, 15[
iii) ]5, 15]
iv) [5, 15]
[II] Complete the following:
1] The sum of the lengths of any two sides in the triangle … greater than.. the length
of the third side.
24
2] If the lengths of two sides in the triangle are unequal then the greater in length is
opposite to … The angle of greater measure…
3] If the measures of two angles in a triangle are unequal, then the greater in
measure is opposite to…. the greater side in length….
4] The longest side in the right – angled triangle is … the hypotenuse ….
5] ∆ ABC is obtuse – angled triangle at A, then the longest side in it is … BC ….
6] ∆ ABC in which mA = 43° and m  B = 47°, then the smallest side in it is … BC …
7] ∆ ABC in which m (A) = 100°, then the greatest side in it is …… BC …..
8] ∆ ABC in which AB = 3 cm, BC = 5 cm, then AC ∈ ]…2. , …8..[
[III] Essay questions:
1] In the opposite figure:
ABCD is a quadrilateral in which
AB is the longest side, CD is the shortest side.
Prove that : m (  C) > m (  A).
In ∆ ABC: AB is the longest side,
AB > BC Þ
m(1) > m(3)
A
43
D
(i)
2
In ∆ ACD: CD is the shortest side
AD > DC
m(2) > m(4)
(ii)
Þ
(i) + (ii) Þ
m(1) + m(2) > m(3) + m(4)
m (  C) > m (  A).
2] In the opposite figure:
ABC is a triangle in which AB > AC,
X ∈ AB , Y  AC , XY // BC .
Prove that:
a) m (  AYX) > m (  AXY).
b) AX > AY
In ABC: AB > AC Þ m(C) > m(B)
(1)
XY // BC
m(1) = m(B) and m(2) = m(C)
(2)
From (1) and (2) Þ m(2) > m(1)
m (  AYX) > m (  AXY).
AX > AY
C
1
B
A
Y
<
X
<
C
B
A
3] In the opposite figure:
AB = AC = CD, m ( BAC) = 40°,
40°
C
/
25
/
D  BC . Prove that : BD > AD
In ∆ ABC: AB = AC , m(BAC) = 40°
/
m(B) = m(ACD) = (180° - 40°)  2 = 70° D
In ∆ ACD: CA = CD Þ m(CAD) = m(D)
m(CAD) = 1/2 m(ACB) = 1/2 ´ 70 = 35° (exterior angle)
m(BAD) = 40° + 35° = 75°
Þ
m(BAD) > m(B)
BD > AD
B