Mathematics
Grade 10
Euclidian Geometry
Question 1:
1.1 In triangle ∆𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴 || 𝐷𝐷𝐷𝐷. Calculate with reasons the sizes of all angles indicated with a
small letter.
𝑧𝑧
4𝑥𝑥
80°
𝑦𝑦
8𝑥𝑥
� 𝐶𝐶 = 20°. Determine the
1.2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a Trapezium with 𝐴𝐴𝐴𝐴 || 𝐷𝐷𝐷𝐷. 𝐷𝐷𝐷𝐷 = 𝐷𝐷𝐷𝐷 and 𝐵𝐵𝐵𝐵 = 𝐵𝐵𝐵𝐵. 𝐴𝐴𝐷𝐷
Unknown angles with reasons.
� 𝑀𝑀 and 𝑂𝑂𝑂𝑂 bisect 𝑁𝑁𝑀𝑀
� 𝑃𝑃.
1.3 In sketch below, 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 is a parallelogram. 𝑂𝑂𝑂𝑂 bisects 𝐾𝐾𝑁𝑁
Let 𝑁𝑁2 = 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑀𝑀2 = 𝑦𝑦, and show that 𝑁𝑁𝑂𝑂�𝑀𝑀 = 90°.
1
Mathematics
Grade 10
Question 2:
2.1
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram. 𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴 and 𝐸𝐸𝐸𝐸 = 𝐷𝐷𝐷𝐷. Determine, with reasons, the values of
𝑥𝑥, 𝑦𝑦 and 𝑝𝑝.
A
𝒙𝒙
A
𝟏𝟏
A
𝒑𝒑
𝟕𝟕𝟕𝟕°
A
A
𝟏𝟏
B
A
2.2
D
𝟐𝟐
E
A
𝒚𝒚
𝟑𝟑
A
A
C
In the diagram below parallelogram 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is given with 𝑄𝑄� = 114°. 𝑃𝑃𝑃𝑃 bisects QPˆ S and 𝑇𝑇𝑇𝑇
bisects PSˆ R . Prove, with reasons, that 𝑃𝑃𝑃𝑃 ⊥ 𝑆𝑆𝑆𝑆.
P
1
2
1
S
2
1
T
Q
2.3
R
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a rhombus. DAˆ F = x and AFˆD = 3x . Prove that 𝐴𝐴𝐴𝐴 bisect DAˆ C .
A
B
2
x1
D
3x
F
1
2
C
2
Mathematics
2.4
Grade 10
The diagram below is not drawn to scale. Given: 𝑄𝑄𝑄𝑄 // 𝑇𝑇𝑇𝑇; 𝑃𝑃𝑃𝑃 // 𝑆𝑆𝑆𝑆. 𝑄𝑄𝑄𝑄 = 𝑇𝑇𝑇𝑇 = 9 cm
and 𝑃𝑃𝑃𝑃 = 15 cm.
P
15
S
V
Q
9
T
R
9
1
2.4.1 Prove that 𝑉𝑉𝑉𝑉 = 7 cm.
2
2.4.2 Calculate 𝑃𝑃𝑃𝑃 if 𝑃𝑃𝑃𝑃 =
Question 3:
3.1
16
𝑉𝑉𝑉𝑉 and hence prove that PQˆ R = 90 o
5
Identify each quadrilateral in the diagrams below:
6.1.1
1.
2.
3.
3.1.1.1 Quad 1
3.1.1.2 Quad 2
3.1.1.3 Quad 3
3.2.1 Calculate the values of 𝑥𝑥 and 𝑦𝑦 in each quadrilateral, with reasons.
3.2.1.1 Quad 1
3.2.1.2 Quad 2
3.2.1.3 Quad 3
3
Mathematics
3.3
Grade 10
Fill in the missing words to complete the following facts concerning quadrilaterals:
The opposite angles of a parallelogram and a rhombus are ____________.
A parallelogram, ____________,
_____________ and
_____________ have two pairs of opposite sides parallel and equal.
Question 4:
4.1 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is a square.
4.1.1 Calculate the length of one side of the square. Round off your answer correct to one
decimal digit.
4.1.2 Calculate the length of 𝑄𝑄𝑄𝑄. Round off your answer correct to one decimal digit.
4.1.3 Calculate the area of the square.
4.2
Determine 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 in the following diagram, with reasons.
4
Mathematics
4.3
Grade 10
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram. 𝐵𝐵𝐵𝐵 // 𝐹𝐹𝐹𝐹. Prove that 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 is a parallelogram. Give reasons for
all statements.
Question 5:
5.1
Consider the diagram below and calculate the unknowns, 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧, giving reasons for your
answers.
5.2
In the diagram below 𝐸𝐸, 𝐹𝐹, 𝐺𝐺 and 𝐻𝐻 are the midpoints of 𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴, 𝐵𝐵𝐵𝐵 and 𝐶𝐶𝐶𝐶 respectively.
𝐸𝐸𝐸𝐸 bisects 𝐷𝐷𝐸𝐸� 𝐺𝐺 and 𝐸𝐸𝐸𝐸 bisects 𝐴𝐴𝐸𝐸� 𝐺𝐺. Prove that:
5
Mathematics
Grade 10
5.2.1 𝐸𝐸𝐸𝐸 // 𝐹𝐹𝐹𝐹 and 𝐹𝐹𝐹𝐹 // 𝐺𝐺𝐺𝐺
5.2.2 𝐸𝐸𝐹𝐹� 𝐺𝐺 = 90°
5.2.3 What kind of quadrilateral is 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸? Give reasons for your answer.
Question 6:
6.1
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a quadrilateral with 𝐴𝐴𝐴𝐴 = 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵. Prove that:
6.1.1 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ≡ 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥
6.1.2 Hence, prove 𝐴𝐴𝐴𝐴 // 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 // 𝐵𝐵𝐵𝐵 (using your answer in 6.1.1). You may NOT
say that it is a parallelogram.
6.2
6.1.3 Now prove 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram using two different reasons.
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a rhombus.
6.2.1 Calculate 𝑥𝑥 and 𝑦𝑦.
6.2.2 Prove that ∆𝐴𝐴𝐴𝐴𝐴𝐴 is equilateral.
6
Mathematics
Grade 10
Question 7:
In the diagram below: 𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄 = 𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆 and 𝑃𝑃𝑃𝑃 // 𝑄𝑄𝑄𝑄 . Prove that:
7.1
7.2
7.3
7.4
7.5
𝑅𝑅�2 = 𝑃𝑃�
∆𝑃𝑃𝑃𝑃𝑃𝑃 ≡ ∆𝑅𝑅𝑅𝑅𝑅𝑅
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is a parallelogram
𝑃𝑃𝑃𝑃 = 3 𝑆𝑆𝑆𝑆
𝑀𝑀𝑅𝑅� 𝑄𝑄 = 90°
Question 8:
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram. Determine the values of 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧.
A
y
D
z
44°
B
x
C
7
Mathematics
Grade 10
Question 9:
9.1
In the sketch 𝐷𝐷𝐷𝐷 ∥ 𝐹𝐹𝐹𝐹 ∥ 𝐵𝐵𝐵𝐵. 𝐴𝐴𝐴𝐴 = 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 = 𝐹𝐹𝐹𝐹. 𝐵𝐵𝐵𝐵 = 160 𝑚𝑚𝑚𝑚.
A
D
E
G
F
C
B
Calculate the length of 𝐷𝐷𝐷𝐷.
9.2 Use the diagram below to answer the questions that follow.
B
B
E
E
CC
h
F
A
D
9.2.1 What is the relationship between 𝐴𝐴𝐴𝐴 and 𝐹𝐹𝐹𝐹?
F
8
Mathematics
Grade 10
9.2.2 Write down a formula for the area of:
i.
ii.
iii.
∆𝐴𝐴𝐴𝐴𝐴𝐴 in terms of 𝐵𝐵𝐵𝐵 and ℎ.
∆𝐴𝐴𝐴𝐴𝐴𝐴 in terms of 𝐴𝐴𝐴𝐴 and ℎ.
Trapezium 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 as a single term.
9.2.3 If 𝐴𝐴𝐴𝐴 = 5 units, 𝐵𝐵𝐵𝐵 = 10 units and ℎ = 4 units, determine the area of 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.
9.3* In Euclidean geometry, a tangent line to a circle is a line that touches the circle at exactly one
point, never entering the circle's interior. The tangent line to a circle at the point of contact,
point A, is perpendicular to the diameter, AB, at that point.
𝑇𝑇𝑇𝑇𝑇𝑇 is a tangent to the circle at 𝐴𝐴.
𝑇𝑇
𝑁𝑁
Determine the equation of the tangent, 𝑇𝑇𝑇𝑇𝑇𝑇, which is a straight line,
𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐. (Hint: First find the gradient of the diameter 𝐴𝐴𝐴𝐴)
9
Mathematics
Grade 10
Question 10:
∆𝑃𝑃𝑃𝑃𝑃𝑃 is an isosceles triangle with 𝑄𝑄𝑄𝑄 = 20 meters and 𝑄𝑄𝑃𝑃�𝑅𝑅 = 105°.
P
105°
Q
20 m
Determine the length of 𝑃𝑃𝑃𝑃. (Hint: Construct the height of ∆𝑃𝑃𝑃𝑃𝑃𝑃)
R
Question11:
20 mm
12 mm
30 mm
38 mm
O is the midpoint of side 𝐴𝐴𝐴𝐴 of 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥. 𝑂𝑂𝑂𝑂 ‖ 𝐵𝐵𝐵𝐵. 𝑂𝑂𝑂𝑂 intersects 𝐴𝐴𝐴𝐴 in 𝐸𝐸. 𝐵𝐵𝐵𝐵 = 38 mm,
𝐴𝐴𝐴𝐴 = 20 mm, 𝐷𝐷𝐷𝐷 = 12 mm and 𝐸𝐸𝐸𝐸 = 30 mm. Give, with reasons, the lengths of:
11.1 𝐷𝐷𝐷𝐷
11.2 𝐴𝐴𝐴𝐴
11.3 𝑂𝑂𝑂𝑂
10
Mathematics
Grade 10
MEMO:
Question 1:
1.1 In triangle ∆𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴 || 𝐷𝐷𝐷𝐷. Calculate with reasons the sizes of all angles indicated with a small
letter.
Answer
Reason
Ext ∠ of ⊿
8𝑥𝑥 = 4𝑥𝑥 + 80°
4𝑥𝑥 = 80°
𝑥𝑥 = 20°
∠ on a straight line
𝑦𝑦 = 180° − 8(20°)
𝑦𝑦 = 180° − 160°
𝑦𝑦 = 20°
Corr ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐷𝐷𝐷𝐷
𝑧𝑧 = 80°
� 𝐶𝐶 = 20°. Determine the
1.2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a Trapezium with 𝐴𝐴𝐴𝐴 || 𝐷𝐷𝐷𝐷. 𝐷𝐷𝐷𝐷 = 𝐷𝐷𝐷𝐷 and 𝐵𝐵𝐵𝐵 = 𝐵𝐵𝐵𝐵. 𝐴𝐴𝐷𝐷
Unknown angles with reasons.
1
Answer
𝑥𝑥 =
180° − 20°
2
𝑥𝑥 = 80°
�1 = 20°
𝐴𝐴
𝑦𝑦 = 20°
𝑧𝑧 = 180° − 40°
Reason
Int ∠ of a ⊿
Opp ∠ s = opp sides
Alt ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐷𝐷𝐷𝐷
Opp ∠ s = opp sides
Int ∠ of a ⊿
11
Mathematics
Grade 10
𝑧𝑧 = 140°
� 𝑀𝑀 and 𝑂𝑂𝑂𝑂 bisects 𝑁𝑁𝑀𝑀
� 𝑃𝑃.
1.3 In sketch below, 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 is a parallelogram. 𝑂𝑂𝑂𝑂 bisects 𝐾𝐾𝑁𝑁
Let 𝑁𝑁2 = 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑀𝑀2 = 𝑦𝑦, show that 𝑁𝑁𝑂𝑂�𝑀𝑀 = 90°.
Answer
𝑁𝑁𝑂𝑂�𝑀𝑀 = 180° − (𝑥𝑥 + 𝑦𝑦)
Reason
Int ∠ of a ⊿
� + 𝑀𝑀
� = 180°
𝑁𝑁
Co int ∠ , 𝐾𝐾𝐾𝐾 ∥ 𝑀𝑀𝑀𝑀
∴ 𝑥𝑥 + 𝑥𝑥 + 𝑦𝑦 + 𝑦𝑦 = 180°
∴ 2𝑥𝑥 + 2𝑦𝑦 = 180°
∴ 𝑥𝑥 + 𝑦𝑦 = 90°
𝑁𝑁𝑂𝑂�𝑀𝑀 = 180° − (90°)
𝑁𝑁𝑂𝑂�𝑀𝑀 = 90°
Question 2:
2.1
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram. 𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴 and 𝐸𝐸𝐸𝐸 = 𝐷𝐷𝐷𝐷. Determine, with reasons, the values of
𝑥𝑥, 𝑦𝑦 and 𝑝𝑝.
A
𝒙𝒙
A
B
𝟏𝟏
A
𝒑𝒑
𝟕𝟕𝟕𝟕°
A
D
A
𝟏𝟏
A
𝟐𝟐
E
A
𝟑𝟑
A
𝒚𝒚
A
C
12
Mathematics
Grade 10
Answer
Reason
𝐸𝐸�1 = 70°
Alt ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
Int ∠ of a ⊿
𝑥𝑥 = 180° − 140° = 40°
Opp ∠ s = opp sides
Opp ∠ of parm
𝑦𝑦 = 70° + 40° = 130°
𝑝𝑝 =
2.2
180°−130°
2
Int ∠ of a ⊿
= 25°
Opp ∠ s = opp sides
In the diagram below parallelogram 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is given with 𝑄𝑄� = 114°. 𝑃𝑃𝑃𝑃 bisects QPˆ S and 𝑇𝑇𝑇𝑇
bisects PSˆ R . Prove, with reasons, that 𝑃𝑃𝑃𝑃 ⊥ 𝑆𝑆𝑆𝑆.
P
1
2
1
S
2
1
T
Q
R
Answer
Reason
𝑄𝑄� = 𝑆𝑆̂ = 114°
Opp ∠ of parm
𝑃𝑃� + 𝑆𝑆̂ = 180°
Co int ∠ , 𝑄𝑄𝑄𝑄 ∥ 𝑆𝑆𝑆𝑆
𝑇𝑇�1 = 180° − 33° − 57° = 90°
Int ∠ of a ⊿
∴ 𝑆𝑆�1 = 57°
�2 =
∴ 𝑃𝑃
180° − 114°
= 33°
2
∴ 𝑃𝑃𝑃𝑃 ⊥ 𝑆𝑆𝑆𝑆
13
Mathematics
2.3
Grade 10
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a rhombus. DAˆ F = x and AFˆD = 3x . Prove that 𝐴𝐴𝐴𝐴 bisect DAˆ C .
A
B
2
x1
D
1
3x
F
2
C
Answer
Reason
𝐴𝐴̂1+2 = 3𝑥𝑥
Alt ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐶𝐶𝐶𝐶
∴ 𝐴𝐴̂ = 4𝑥𝑥
𝐴𝐴̂1 = 2𝑥𝑥 − 𝑥𝑥 = 𝑥𝑥
∴ 𝐴𝐴𝐴𝐴 bisect 𝐷𝐷𝐴𝐴̂𝐶𝐶.
2.4
The diagram below is not drawn to scale. Given: 𝑄𝑄𝑄𝑄 // 𝑇𝑇𝑇𝑇; 𝑃𝑃𝑃𝑃 // 𝑆𝑆𝑆𝑆. 𝑄𝑄𝑄𝑄 = 𝑇𝑇𝑇𝑇 = 9 cm
and 𝑃𝑃𝑃𝑃 = 15 cm.
P
15
S
V
Q
9
T
1
2.4.1 Prove that 𝑉𝑉𝑉𝑉 = 7 cm.
2
Answer
𝑆𝑆𝑆𝑆 = 15cm
𝑆𝑆𝑆𝑆 = 𝑉𝑉𝑉𝑉
9
R
Reason
Midpoint theorem
Midpoint theorem
1
∴ 𝑉𝑉𝑉𝑉 = 7 2cm
14
Mathematics
Grade 10
2.4.2 Calculate 𝑃𝑃𝑃𝑃 if 𝑃𝑃𝑃𝑃 =
Answer
𝑃𝑃𝑃𝑃 =
16
𝑉𝑉𝑉𝑉 and hence prove that PQˆ R = 90 o
5
Reason
16 15
5
� 2 � = 24 cm
𝑃𝑃𝑃𝑃 2 = 𝑄𝑄𝑄𝑄 2 + 𝑃𝑃𝑃𝑃 2
Theorem of Pythagoras.
302 = 182 + 242
900 = 900
∴ 𝑃𝑃𝑄𝑄� 𝑅𝑅 = 90°
Question 3:
3.1
Identify each quadrilateral in the diagrams below:
6.1.1
1.
2.
3.
3.1.1.1 Quad 1
Trapezium
3.1.1.2 Quad 2
Rhombus
3.1.1.3 Quad 3
Parallelogram
15
Mathematics
Grade 10
3.2.2 Calculate the values of 𝑥𝑥 and 𝑦𝑦 in each quadrilateral, with reasons.
3.2.1.1 Quad 1
Answer
Reason
Co int ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐶𝐶𝐶𝐶
𝑥𝑥 + 30° + 𝑥𝑥 − 10° = 180°
2𝑥𝑥 = 160°
𝑥𝑥 = 80°
3.2.1.2 Quad 2
Answer
Reason
𝑥𝑥 = 40°
Opp ∠ s = opp sides
𝑦𝑦 = 50°
Alt int ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
Diagonal bisect perpendicularly
𝑦𝑦 = 180° − 90° − 40°
Int ∠ of a ⊿
3.2.1.3 Quad 3
Answer
𝑥𝑥 = 62°
𝑦𝑦 = 15cm
3.3
Reason
Alt int ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
Diagonal bisect
Fill in the missing words to complete the following facts concerning quadrilaterals:
The opposite angles of a parallelogram and a rhombus are equal.
A parallelogram, rhombus, square and rectangle have two pairs of opposite sides
parallel and equal.
16
Mathematics
Grade 10
Question 4:
4.1
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is a square.
4.1.1 Calculate the length of one side of the square. Round off your answer correct to one
decimal digit.
Answer
𝑄𝑄𝑄𝑄: cos 37° =
≈ 8 mm
Reason
𝑄𝑄𝑄𝑄
= 7.98 …
10
4.1.2 Calculate the length of 𝑄𝑄𝑄𝑄. Round off your answer correct to one decimal digit.
Answer
𝑄𝑄𝑄𝑄: sin 42° =
≈ 12mm
Reason
8
= 11.95 …
𝑄𝑄𝑄𝑄
4.1.3 Calculate the area of the square.
Answer
Reason
𝐴𝐴 = 𝑙𝑙 2
𝐴𝐴 = 82
𝐴𝐴 = 64 mm2
17
Mathematics
4.2
Grade 10
Determine 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 in the following diagram, with reasons.
Answer
Reason
𝐷𝐷𝐷𝐷 ∥ 𝐴𝐴𝐴𝐴
Midpoint theorem
∴ 𝑥𝑥 = 90° − 32°
Corresp ∠ , 𝐷𝐷𝐷𝐷 ∥ 𝐴𝐴𝐴𝐴
𝑦𝑦 = 58°
Corresp ∠ , 𝐷𝐷𝐷𝐷 ∥ 𝐴𝐴𝐴𝐴
𝑥𝑥 = 58°
Int ∠ of a ⊿
1
𝑧𝑧 = 𝐴𝐴𝐴𝐴
2
Midpoint theorem
𝑧𝑧 = 12,7 cm
4.3
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram. 𝐵𝐵𝐵𝐵 // 𝐹𝐹𝐹𝐹. Prove that 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 is a parallelogram. Give reasons for
all statements.
Answer
Reason
In ⊿ 𝐴𝐴𝐴𝐴𝐴𝐴 and ⊿𝐶𝐶𝐶𝐶𝐶𝐶:
Opp ∠ of parm
𝐴𝐴𝐴𝐴 = 𝐶𝐶𝐶𝐶
Opp sides of parm
𝐴𝐴𝐸𝐸� 𝐵𝐵 = 𝐸𝐸𝐵𝐵� 𝐹𝐹
Alt ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
𝐴𝐴̂ = 𝐶𝐶̂
𝐸𝐸𝐵𝐵� 𝐹𝐹 = 𝐴𝐴𝐸𝐸� 𝐵𝐵
∴ 𝐴𝐴𝐸𝐸� 𝐵𝐵 = 𝐴𝐴𝐸𝐸� 𝐵𝐵
Corre ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
∴In ⊿ 𝐴𝐴𝐴𝐴𝐴𝐴 ≡ ⊿𝐶𝐶𝐶𝐶𝐶𝐶
∠∠ S
𝐸𝐸𝐸𝐸 = 𝐵𝐵𝐵𝐵
Opp sides are equal
∴ 𝐵𝐵𝐵𝐵 = 𝐹𝐹𝐹𝐹
∴ 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 is a parm
Proven
18
Mathematics
Grade 10
Question 5:
5.1
Consider the diagram below and calculate the unknowns, 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧, giving reasons for your
answers.
Answer
Reason
𝐷𝐷𝐷𝐷 ∥ 𝐵𝐵𝐵𝐵
Midpoint theorem
𝑥𝑥 = 180° − 60° − 70°
Int ∠ of a ⊿
𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵 and 𝐴𝐴𝐴𝐴 = 𝐸𝐸𝐸𝐸
𝑥𝑥 = 50°
Corre ∠ , 𝐷𝐷𝐷𝐷 ∥ 𝐵𝐵𝐵𝐵
𝑥𝑥 = 𝑦𝑦 = 50°
Midpoint theorem
𝑧𝑧 = 2𝐷𝐷𝐷𝐷
𝑧𝑧 = 4cm
5.2
In the diagram below 𝐸𝐸, 𝐹𝐹, 𝐺𝐺 and 𝐻𝐻 are the midpoints of 𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴, 𝐵𝐵𝐵𝐵 and 𝐶𝐶𝐶𝐶 respectively.
𝐸𝐸𝐸𝐸 bisects 𝐷𝐷𝐸𝐸� 𝐺𝐺 and 𝐸𝐸𝐸𝐸 bisects 𝐴𝐴𝐸𝐸� 𝐺𝐺. Prove that:
5.2.1 𝐸𝐸𝐸𝐸 // 𝐹𝐹𝐹𝐹 and 𝐹𝐹𝐹𝐹 // 𝐺𝐺𝐺𝐺
Answer
Reason
𝐵𝐵𝐵𝐵 = 𝐺𝐺𝐺𝐺 and 𝐵𝐵𝐵𝐵 = 𝐹𝐹𝐹𝐹
Given
𝐷𝐷𝐷𝐷 = 𝐸𝐸𝐸𝐸 and 𝐷𝐷𝐷𝐷 = 𝐻𝐻𝐻𝐻
Given
∴ 𝐴𝐴𝐴𝐴 // 𝐹𝐹𝐹𝐹
Midpoint theorem
∴ 𝐸𝐸𝐸𝐸 // 𝐴𝐴𝐴𝐴
Midpoint theorem
∴ 𝐸𝐸𝐸𝐸 ∥ 𝐹𝐹𝐹𝐹
19
Mathematics
Grade 10
5.2.2 𝐸𝐸𝐹𝐹� 𝐺𝐺 = 90°
Answer
�1 + 𝐸𝐸
�2 + 𝐸𝐸
�3 + 𝐸𝐸
�4 = 180°
𝐸𝐸
Reason
∠ on a straight-line
∴ 𝑥𝑥 + 𝑥𝑥 + 𝑦𝑦 + 𝑦𝑦 = 180°
∴ 2𝑥𝑥 + 2𝑦𝑦 = 180°
∴ 𝑥𝑥 + 𝑦𝑦 = 90°
�2 + 𝐸𝐸
�3 = 180°
𝐸𝐸𝐹𝐹� 𝐺𝐺 + 𝐸𝐸
Co Int ∠ 𝐸𝐸𝐸𝐸 ∥ 𝐹𝐹𝐹𝐹
∴ 𝐸𝐸𝐹𝐹� 𝐺𝐺 + 𝑥𝑥 + 𝑦𝑦 = 180°
∴ 𝐸𝐸𝐹𝐹� 𝐺𝐺 = 90°
5.2.3 What kind of quadrilateral is 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸? Give reasons for your answer.
Answer
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 is a rectangle
Reason
A parm with one 90° angle
Question 6:
6.1
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a quadrilateral with 𝐴𝐴𝐴𝐴 = 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵. Prove that:
6.1.1 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ≡ 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥
Answer
In 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 and 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥:
Reason
𝐴𝐴𝐴𝐴 = 𝐶𝐶𝐶𝐶
Given
𝐷𝐷𝐷𝐷 = 𝐷𝐷𝐷𝐷
Common side
𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵
Given
∴ 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ≡ 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥
SSS
20
Mathematics
Grade 10
6.1.2 Hence, prove 𝐴𝐴𝐴𝐴 // 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 // 𝐵𝐵𝐵𝐵 (using your answer in 6.1.1). You may NOT
say that it is a parallelogram.
Answer
Reason
�2 = 𝐵𝐵
�1
𝐷𝐷
𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ≡ 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥
�2 = 𝐷𝐷
�1
𝐵𝐵
𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ≡ 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥
Alt ∠ are equal
∴ 𝐴𝐴𝐴𝐴 ∥ 𝐷𝐷𝐷𝐷
Alt ∠ are equal
∴ 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
6.1.3 Now prove 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram using two different reasons.
Answer
Reason
𝐴𝐴𝐴𝐴 = 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵
Proven
∴ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parm
2 pairs of opposite sides are equal and
𝐴𝐴𝐴𝐴 ∥ 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 ∥ 𝐵𝐵𝐵𝐵
6.2
Proven
parallel
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a rhombus.
6.2.1 Calculate 𝑥𝑥 and 𝑦𝑦.
Answer
Reason
� = 𝐵𝐵�
𝐷𝐷
Opp ∠ of rhombus
� = 360°
𝐴𝐴̂ + 𝐵𝐵� + 𝐶𝐶̂ + 𝐷𝐷
Int ∠ of quad
𝑥𝑥 = 2𝑦𝑦
3𝑥𝑥
𝑥𝑥 + 2𝑦𝑦 + 2 � − 2𝑦𝑦� = 360°
2
6𝑥𝑥
𝑥𝑥 + 2𝑦𝑦 +
− 4𝑦𝑦 = 360°
2
Opp ∠ of rhombus
2𝑥𝑥 + 4𝑦𝑦 + 6𝑥𝑥 − 8𝑦𝑦 = 720°
8𝑥𝑥 − 4𝑦𝑦 = 720°
8(2𝑦𝑦) − 4𝑦𝑦 = 720°
16𝑦𝑦 − 4𝑦𝑦 = 720°
21
Mathematics
Grade 10
12𝑦𝑦 = 720°
𝑦𝑦 = 60°
Proven
𝑥𝑥 = 2𝑦𝑦
∴ 𝑥𝑥 = 2(60°)
∴ 𝑥𝑥 = 120°
6.2.2 Prove that ∆𝐴𝐴𝐴𝐴𝐴𝐴 is equilateral.
Answer
3(120°)
− 2(60°) = 60°
2
120°
𝐷𝐷𝐴𝐴̂𝐶𝐶 = 𝐷𝐷𝐶𝐶̂ 𝐴𝐴 =
= 60°
2
�=
𝐷𝐷
∴ ∆𝐴𝐴𝐴𝐴𝐴𝐴 is equilateral
Reason
Diag of rhombus bisect ∠
All ∠𝑠𝑠 equal to 60°
Question 7:
In the diagram below: 𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄 = 𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆 and 𝑃𝑃𝑃𝑃 // 𝑄𝑄𝑄𝑄 . Prove that:
7.1
𝑅𝑅�2 = 𝑃𝑃�
Answer
Reason
�1
�2 = 𝑄𝑄
𝑅𝑅
Opp ∠ equals opp sides
𝑆𝑆�1 = 𝑃𝑃�
Opp ∠ equals opp sides
�1
𝑆𝑆�1 = 𝑄𝑄
Alt ∠ , 𝑃𝑃𝑃𝑃 ∥ 𝑄𝑄𝑄𝑄
∴ 𝑅𝑅�2 = 𝑃𝑃�
7.2
∆𝑃𝑃𝑃𝑃𝑃𝑃 ≡ ∆𝑅𝑅𝑅𝑅𝑅𝑅
Answer
Reason
𝑅𝑅�2 = 𝑃𝑃�
Proven
𝑄𝑄𝑄𝑄 = 𝑄𝑄𝑄𝑄
Common side
�1
𝑆𝑆�1 = 𝑄𝑄
Proven
∴ ∆𝑃𝑃𝑃𝑃𝑃𝑃 ≡ ∆𝑅𝑅𝑅𝑅𝑅𝑅
∠∠S
22
Mathematics
7.3
Grade 10
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is a parallelogram
Answer
Reason
𝑃𝑃𝑃𝑃 = 𝑅𝑅𝑅𝑅
Given
∴ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is a parm
2 Pairs of opp sides equal and parallel
Proven
𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄
7.4
𝑃𝑃𝑃𝑃 = 3 𝑆𝑆𝑆𝑆
Answer
Reason
𝑀𝑀𝑀𝑀 = 𝑇𝑇𝑇𝑇
𝑆𝑆𝑆𝑆 ∥ 𝑄𝑄𝑄𝑄 and 𝑀𝑀𝑀𝑀 = 𝑆𝑆𝑆𝑆
𝑄𝑄𝑄𝑄 = 𝑃𝑃𝑃𝑃
Opp sides of parm
1
∴ 𝑆𝑆𝑆𝑆 = 𝑄𝑄𝑄𝑄
2
Midpoint theorem
1
∴ 𝑆𝑆𝑆𝑆 = 𝑃𝑃𝑃𝑃
2
∴ 𝑃𝑃𝑃𝑃 = 3𝑆𝑆𝑆𝑆
7.5
𝑀𝑀𝑅𝑅� 𝑄𝑄 = 90°
Answer
Reason
�
𝑅𝑅�1 = 𝑀𝑀
Opp ∠ equals opp sides
But 𝑅𝑅�2 = 𝑄𝑄�1
Opp ∠ equals opp sides
� + 𝑅𝑅�1 + 𝑅𝑅�𝑠𝑠 + 𝑄𝑄�1 = 180°
𝑀𝑀
Int ∠𝑠𝑠 of a triangle
∴ 2𝑅𝑅�1 + 2𝑅𝑅�2 = 180°
2�𝑅𝑅�1 + 𝑅𝑅�2 � = 180°
𝑅𝑅�1 + 𝑅𝑅�2 = 90°
∴ 𝑀𝑀𝑅𝑅� 𝑄𝑄 = 90°
Question 8:
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a parallelogram. Determine the values of 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧.
y
D
A
z
44°
x
C
23
B
Mathematics
Grade 10
Answer
Reason
𝑥𝑥 = 44°
Opp sides of parm ∥
𝑥𝑥 = 𝑦𝑦 = 44°
Opp ∠ equals opp sides
Alt ∠ , 𝐴𝐴𝐴𝐴 ∥ 𝐷𝐷𝐷𝐷
Opp ∠ of parm are equal
𝑦𝑦 = 𝑧𝑧 = 44°
A
Question 9:
9.1
In the sketch 𝐷𝐷𝐷𝐷 ∥ 𝐹𝐹𝐹𝐹 ∥ 𝐵𝐵𝐵𝐵. 𝐴𝐴𝐴𝐴 = 𝐷𝐷𝐷𝐷 and 𝐴𝐴𝐴𝐴 = 𝐹𝐹𝐹𝐹. 𝐵𝐵𝐵𝐵 = 160 𝑚𝑚𝑚𝑚.
D
G
F
Calculate the length of 𝐷𝐷𝐷𝐷.
B
Answer
1
E
C
Reason
Midpoint theorem
𝐹𝐹𝐹𝐹 = 2 𝐵𝐵𝐵𝐵 = 80mm
𝐷𝐷𝐷𝐷 ∥ 𝐹𝐹𝐹𝐹 ∥ 𝐵𝐵𝐵𝐵 and 𝐴𝐴𝐴𝐴 = 𝐷𝐷𝐷𝐷, 𝐴𝐴𝐴𝐴 =
𝐹𝐹𝐹𝐹
1
Midpoint theorem
𝐷𝐷𝐷𝐷 = 2 𝐹𝐹𝐹𝐹 = 40mm
9.2 Use the diagram below to answer the questions that follow.
B
E
C
h
9.2.1 What is the relationship between 𝐴𝐴𝐴𝐴 and 𝐹𝐹𝐹𝐹?
Answer
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 is a rectangle
∴ 𝐴𝐴𝐴𝐴 = 𝐹𝐹𝐹𝐹
A
D
F
Reason
Opp sides parallel and ∠ = 90°
Opp sides of rectangle
24
Mathematics
Grade 10
9.2.2 Write down a formula for the area of:
iv.
v.
vi.
∆𝐴𝐴𝐴𝐴𝐴𝐴 in terms of 𝐵𝐵𝐵𝐵 and ℎ.
∆𝐴𝐴𝐴𝐴𝐴𝐴 in terms of 𝐴𝐴𝐴𝐴 and ℎ.
𝐴𝐴 =
𝐴𝐴 =
Trapezium 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 as a single term. 𝐴𝐴 =
𝐵𝐵𝐵𝐵
2
𝐴𝐴𝐴𝐴
2
.ℎ
.ℎ
ℎ(𝐵𝐵𝐵𝐵+𝐴𝐴𝐴𝐴)
2
9.2.3 If 𝐴𝐴𝐴𝐴 = 5 units, 𝐵𝐵𝐵𝐵 = 10 units and ℎ = 4 units, determine the area of 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.
Answer
𝐴𝐴 =
𝐴𝐴 =
Reason
ℎ(𝐵𝐵𝐵𝐵 + 𝐴𝐴𝐴𝐴)
2
4(10+5)
2
= 30 square units
9.3* In Euclidean geometry, a tangent line to a circle is a line that touches the circle at exactly one
point, never entering the circle's interior. The tangent line to a circle at the point of contact,
point A, is perpendicular to the diameter, AB, at that point.
𝑇𝑇𝑇𝑇𝑇𝑇 is a tangent to the circle at 𝐴𝐴.
𝑇𝑇
N
Determine the equation of the tangent, 𝑇𝑇𝑇𝑇𝑇𝑇, which is a straight line,
𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐. (Hint: First find the gradient of the diameter 𝐴𝐴𝐴𝐴)
𝑦𝑦𝐵𝐵 − 𝑦𝑦𝐴𝐴
𝐴𝐴𝐴𝐴 =
𝑥𝑥𝐵𝐵 − 𝑥𝑥𝐴𝐴
𝐴𝐴𝐴𝐴 =
5−3
−1
=
−1 − 5
3
𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇 = 3
𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐
3 = 3(5) + 𝑐𝑐
25
Mathematics
Grade 10
3 = 15 + 𝑐𝑐
𝑐𝑐 = −12
𝑦𝑦 = 3𝑥𝑥 − 12
Question 10:
∆𝑃𝑃𝑃𝑃𝑃𝑃 is an isosceles triangle with 𝑄𝑄𝑄𝑄 = 20 meters and 𝑄𝑄𝑃𝑃�𝑅𝑅 = 105°.
P
105°
S
Q
20 m
Determine the length of 𝑃𝑃𝑃𝑃. (Hint: Construct the height of ∆𝑃𝑃𝑃𝑃𝑃𝑃)
Answer
R
Reason
Construct the height of ∆𝑃𝑃𝑃𝑃𝑃𝑃
𝑄𝑄𝑄𝑄 = 𝑆𝑆𝑆𝑆 = 10m
𝑄𝑄� =
180° − 105°
= 37,5°
2
Int ∠𝑠𝑠 of a triangle
10
𝑄𝑄𝑄𝑄: cos 37.5° = 𝑃𝑃𝑃𝑃 = 12,6 m
Question 11:
20 mm
12 mm
30 mm
38 mm
O is the midpoint of side 𝐴𝐴𝐴𝐴 of 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥. 𝑂𝑂𝑂𝑂 ‖ 𝐵𝐵𝐵𝐵. 𝑂𝑂𝑂𝑂 intersects 𝐴𝐴𝐴𝐴 in 𝐸𝐸. 𝐵𝐵𝐵𝐵 = 38 mm,
𝐴𝐴𝐴𝐴 = 20 mm, 𝐷𝐷𝐷𝐷 = 12 mm and 𝐸𝐸𝐸𝐸 = 30 mm. Give, with reasons, the lengths of:
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Mathematics
Grade 10
11.1 𝐷𝐷𝐷𝐷
11.2 𝐴𝐴𝐴𝐴
11.3 𝑂𝑂𝑂𝑂
Answer
Reason
𝐷𝐷𝐷𝐷 = 𝐴𝐴𝐴𝐴 = 20mm
𝐴𝐴𝐴𝐴 ∥ 𝑂𝑂𝑂𝑂 and 𝐴𝐴𝐴𝐴 = 𝑂𝑂𝑂𝑂
𝐴𝐴𝐴𝐴 = 2𝐸𝐸𝐸𝐸
Midpoint theorem
∴ 𝐴𝐴𝐴𝐴 = 60mm
𝐶𝐶𝐶𝐶 = 2𝐷𝐷𝐷𝐷
Midpoint theorem
∴ 𝐶𝐶𝐶𝐶 = 24mm
1
𝑂𝑂𝑂𝑂 = 𝐵𝐵𝐵𝐵
2
Midpoint theorem
𝑂𝑂𝑂𝑂 =
38+24
2
Midpoint theorem
= 31mm
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