Post on 31-Dec-2015
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Geometry
Unit VIAreas of Triangles, Parallelograms, Trapezoids, Rhombuses and Kites
Area of a Parallelogram
Draw and BE CF perpendicular to AD�������������� �
. Note that ABE DCF . If these two triangles are congruent, then their areas are equal. Consider cutting off ABE and placing it on top of DCF . You can see that the area of parallelogram ABCD is equal to the area of rectangle EBCF. Theorem 6.21: For a parallelogram with base b and height h, the area is given by the formula __b*h___ Note that the height (also called the altitude) is the length of the segment ___perpendicular to the base
from a point on the opposite side_____________________.
A
B C
D E F
Example: Find the area of parallelogram ABCD. Example: Find the value of x. Quad. EFGH is a parallelogram.
A
8
15 D C
B
60°
E
6 8
10
x
H G
F
30
2𝑠=8𝑠=4
𝑠√3=4 √3
𝐴= h𝑏𝐴=15 (4√3)𝐴=60√3𝑢𝑛𝑖𝑡 𝑠2
𝑎𝑟𝑒𝑎𝑜𝑓 𝐸𝐹𝐺𝐻=6∗10𝑎𝑟𝑒𝑎𝑜𝑓 𝐸𝐹𝐺𝐻=6060=8∗ 𝑥𝑥=7.5
Area of a Triangle Any triangle is half of a parallelogram. Theorem 6.22: For a triangle with base b and height h, the area is given by the formula
h
Example: Find the area of ABC to the 1000th. Example: Find the area of an isosceles right triangle that has a hypotenuse of length 20 cm.
10
C
A
B 25
10 y
x
cos 25=𝑥
10
10∗ cos25=𝑥9 .063=𝑥
sin 25=𝑦
10
4 .226=𝑦10∗ sin 25=𝑦
𝐴=12
h𝑏
𝐴=12∗9.063∗4.226
𝐴=19.150𝑢𝑛𝑖𝑡 𝑠2
20
20=𝑎√2𝑎=10 √2
𝑎=10 √2
𝐴=12
h𝑏
𝐴=12∗10√2∗10√2
𝐴=100𝑐𝑚2
Example: Find the area of DEF with vertices D(–1 , –6), E(-1, 3) and F(2, 0).
𝐴𝑅𝐸𝐴𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒=12
h𝑏9
𝑏𝑎𝑠𝑒=9h h𝑒𝑖𝑔 𝑡=3
𝐴𝑅𝐸𝐴𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒=12∗9∗3
𝐴𝑅𝐸𝐴𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒=13.5𝑢𝑛𝑖𝑡 𝑠2
Area of a Trapezoid
Draw diagonal XZ . WXZA __________ XYZA __________
So Trap WXYZA __________+__________ = ____________________
Theorem 6.23: For a trapezoid with bases b1 and b2 and height h, the area of a trapezoid is given by the
formula ___________________
Y X
W Z
h
b2
b1
Example: A trapezoid has an area of 108.8 in2 and bases of lengths 12in. and 20in. Find the height of the trapezoid.
𝐴=12
h(𝑏1+𝑏2)
108.8=12
h(12+20)
108.8=12
h(32)
217.6=32 h
6.8=h
12
20
h
Area of Rhombuses and Kites Recall that the diagonals of both rhombuses and kites are __perpendicular_______. Area ABD = ____________ Area BCD = ______________ Theorems 6.24 & 6.25: For a rhombus or kite with diagonals d1 and d2, the
area is given by the formula
D
A
B
C
Example 1: BD= 12 and AC= 18. Find the area of the kite. . Example 2: BD=13.5 and AC= 21. Find the area of the kite.
D
A
B
C
D
A
B
C
𝐴=12(𝑑1)(𝑑2)
𝐴=12(𝑑1)(𝑑2)
12
18𝐴=
12(12)(18)
𝐴=108
13.5
21
𝐴=12(13.5)(21)
𝐴=141.75
Area Formulas
• Area of a triangle:• Area of parallelogram:• Area of a kite:• Area of a rhombus:• Area of a rectangle:– Perimeter of a rectangle:
• Area of a square:– Perimeter of a square:
𝐴=12
h𝑏
𝐴= h𝑏
𝐴=12𝑑1𝑑2
𝐴=12𝑑1𝑑2
𝐴=𝑙∗𝑤𝑃=2𝑙+2𝑤
𝐴=𝑠2
𝑃=4 𝑠