Buksis-8
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8 .3
What are you going to learn? Form of the windows trellis on the left figure.
To describe the properties of a parallelogram To define a parallelogram To discover the formula of the perimeter and area of a parallelogram
Key Term:
Look at the figure above.
• parallelogram
What do you think about the shape of the windows trellis above? The quadrilaterals on the figure above are called parallelograms. Those parallelograms have equal forms and sizes. Now, look at the following parallelogram and discuss with your classmates how a parallelogram can be formed by a rectangle.
(i)
(ii)
(iii)
Figure 8.2
Figure 8.2 shows that a parallelogram can be formed by cutting the rectangle (Figure 8.2(ii)) and slide the cut to the right to form a parallelogram (Figure 8.2(iii)). To make sure, do activities (i), (ii), and (iii) in Figure 8.2 above using a piece of paper.
Mathematics for Junior High School Year 7 / 319
Properties of parallelogram: A
B
1. The opposite sides are equal and parallel;
O D
C Figure 8.3
AB // CD , AD // BC , AB = DC, and AD = BC.
Why? Explain.
2. The opposite angles are equal, namely ∠A = ∠C and ∠B = ∠D. Why? Explain. 3. Two adjacent angles are supplementary, namely ∠A+∠B = ∠B+ ∠ C = ∠C+∠D = ∠D+ ∠ A = 180 0 . 4. The diagonal bisects the area of the parallelogram, that is the area of ΔACB = the area ΔCAD and the area ΔADB = the area ΔCBD. 5. The diagonals of a parallelogram bisect each other, that is AO = CO and BO = DO. After you know the properties of a parallelogram, what can you say about parallelogram?
Parallelogram
A quadrilateral whose opposite sides are parallel.
Now you will find out the formula of the area and perimeter of a parallelogram.
Mini - Lab WORK IN GROUPS Material: grid paper, a pencil, and a scissors. On a grid paper, draw and cut a parallelogram. Draw a height line of the parallelogram and then cut it into two parts along the dotted line. Combine these two parts to form a rectangle. Discuss with your classmates to answer the following questions.
Compare the area of a rectangle and the area of the original parallelogram. What do you get? Are the heights of a rectangle and parallelogram equal? Are the bases of a rectangle and parallelogram equal? In your own words, express the formula of the perimeter and the area of a parallelogram.
320 / Student’s Book – Quadrilaterals
height tin
∟
base
Area and Perimeter of Parallelogram
Height=t
b
Base = a
The area of parallelogram is the product of the base and the height. • The perimeter of a parallelogram is twice of two adjacent sides of the parallelogram. •
If a parallelogram has area A, base a, adjacent side of a is b and height t, then A=a×t P = 2 (a + b)
D
Calculate the area of parallelogram ABCD on the right figure. A
Solution: Given
10m
C
8m B
: AB = 10 m and height = 8 m
Question : The area of parallelogram ABCD Solution
: Suppose the area of parallelogram is A m2, then A = 10 × 8 A = 80 So, the area of parallelogram ABCD is 80 m2.
Mathematics for Junior High School Year 7 / 321
1. Let EFGH be a parallelogram. Are the following statements true or false? Give your reasons if the statements are true. a. FE // GH G
F
b. Area of ΔFHE = area of ΔHFG D E
H
2.
c. ∠FEH = ∠HGF d. FD = DG 1 EG e. DE = 2
R
Let RSTU be a parallelogram and ∠RST = 80°, calculate ∠SRU and S
∠TUR.
U
80°
T
3. On quadrilateral ABCD, AB // DC , ∠ABC = 125° and ∠BAD = 55°. Is quadrilateral ABCD a parallelogram? Give your reasons. 4. On quadrilateral ABCD, ∠ABC=60 0 , ∠BCD = 120°, ∠CDA = 65°, and ∠DAB = 115°. Is quadrilateral parallelogram? Give your reasons.
ABCD
a
5. KLMN is a parallelogram with diagonals KM and NL intersecting at point P. If KP = 4a+5, KM = 13a, and PL = a+8, what is the length of PN? 6. ABCD is a parallelogram as in the right side figure, then calculate the area of
12
D 6
C
10
ABCD, the length CF and the perimeter A of ABCD.
B
∟ F
7. ABCD is a parallelogram with base AB = 12 cm and the height of AB is 4 cm, then show parallelogram ABCD is 48 cm 2 . 8. Critical
Thinking.
parallelogram if:
322 / Student’s Book – Quadrilaterals
What
that the area of a
happens
with
a
new
a. Its height is twice the height of the original parallelogram? b. Its base and height are twice the base and height of the original parallelogram?
Mathematics for Junior High School Year 7 / 323
324 / Student’s Book – Quadrilaterals
What are you going to learn? Form of the windows trellis on the left figure.
To describe the properties of a parallelogram To define a parallelogram To discover the formula of the perimeter and area of a parallelogram
Key Term:
Look at the figure above.
• parallelogram
What do you think about the shape of the windows trellis above? The quadrilaterals on the figure above are called parallelograms. Those parallelograms have equal forms and sizes. Now, look at the following parallelogram and discuss with your classmates how a parallelogram can be formed by a rectangle.
(i)
(ii)
(iii)
Figure 8.2
Figure 8.2 shows that a parallelogram can be formed by cutting the rectangle (Figure 8.2(ii)) and slide the cut to the right to form a parallelogram (Figure 8.2(iii)). To make sure, do activities (i), (ii), and (iii) in Figure 8.2 above using a piece of paper.
Mathematics for Junior High School Year 7 / 319
Properties of parallelogram: A
B
1. The opposite sides are equal and parallel;
O D
C Figure 8.3
AB // CD , AD // BC , AB = DC, and AD = BC.
Why? Explain.
2. The opposite angles are equal, namely ∠A = ∠C and ∠B = ∠D. Why? Explain. 3. Two adjacent angles are supplementary, namely ∠A+∠B = ∠B+ ∠ C = ∠C+∠D = ∠D+ ∠ A = 180 0 . 4. The diagonal bisects the area of the parallelogram, that is the area of ΔACB = the area ΔCAD and the area ΔADB = the area ΔCBD. 5. The diagonals of a parallelogram bisect each other, that is AO = CO and BO = DO. After you know the properties of a parallelogram, what can you say about parallelogram?
Parallelogram
A quadrilateral whose opposite sides are parallel.
Now you will find out the formula of the area and perimeter of a parallelogram.
Mini - Lab WORK IN GROUPS Material: grid paper, a pencil, and a scissors. On a grid paper, draw and cut a parallelogram. Draw a height line of the parallelogram and then cut it into two parts along the dotted line. Combine these two parts to form a rectangle. Discuss with your classmates to answer the following questions.
Compare the area of a rectangle and the area of the original parallelogram. What do you get? Are the heights of a rectangle and parallelogram equal? Are the bases of a rectangle and parallelogram equal? In your own words, express the formula of the perimeter and the area of a parallelogram.
320 / Student’s Book – Quadrilaterals
height tin
∟
base
Area and Perimeter of Parallelogram
Height=t
b
Base = a
The area of parallelogram is the product of the base and the height. • The perimeter of a parallelogram is twice of two adjacent sides of the parallelogram. •
If a parallelogram has area A, base a, adjacent side of a is b and height t, then A=a×t P = 2 (a + b)
D
Calculate the area of parallelogram ABCD on the right figure. A
Solution: Given
10m
C
8m B
: AB = 10 m and height = 8 m
Question : The area of parallelogram ABCD Solution
: Suppose the area of parallelogram is A m2, then A = 10 × 8 A = 80 So, the area of parallelogram ABCD is 80 m2.
Mathematics for Junior High School Year 7 / 321
1. Let EFGH be a parallelogram. Are the following statements true or false? Give your reasons if the statements are true. a. FE // GH G
F
b. Area of ΔFHE = area of ΔHFG D E
H
2.
c. ∠FEH = ∠HGF d. FD = DG 1 EG e. DE = 2
R
Let RSTU be a parallelogram and ∠RST = 80°, calculate ∠SRU and S
∠TUR.
U
80°
T
3. On quadrilateral ABCD, AB // DC , ∠ABC = 125° and ∠BAD = 55°. Is quadrilateral ABCD a parallelogram? Give your reasons. 4. On quadrilateral ABCD, ∠ABC=60 0 , ∠BCD = 120°, ∠CDA = 65°, and ∠DAB = 115°. Is quadrilateral parallelogram? Give your reasons.
ABCD
a
5. KLMN is a parallelogram with diagonals KM and NL intersecting at point P. If KP = 4a+5, KM = 13a, and PL = a+8, what is the length of PN? 6. ABCD is a parallelogram as in the right side figure, then calculate the area of
12
D 6
C
10
ABCD, the length CF and the perimeter A of ABCD.
B
∟ F
7. ABCD is a parallelogram with base AB = 12 cm and the height of AB is 4 cm, then show parallelogram ABCD is 48 cm 2 . 8. Critical
Thinking.
parallelogram if:
322 / Student’s Book – Quadrilaterals
What
that the area of a
happens
with
a
new
a. Its height is twice the height of the original parallelogram? b. Its base and height are twice the base and height of the original parallelogram?
Mathematics for Junior High School Year 7 / 323
324 / Student’s Book – Quadrilaterals
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