Geometry ReviewAugust 2011
5
7
10
𝑥
12
𝑥+10 512
=10𝑥+10
5 (𝑥+10 )=1205 𝑥+50=120
5 𝑥=705𝑥5
=705
𝑥=14
P
Q
RS
TP
Q
RS
T
7
25
𝑎2+𝑏2=𝑐2
72+𝑏2=252
49+𝑏2=625𝑏2=576
√𝑏2=√576𝑏=24
𝑑=√ (𝑥2−𝑥1 )2+( 𝑦2− 𝑦1 )2
𝑑=√ (9−1 )2+(2−−4 )2
𝑑=√ (8 )2+ (6 )2
𝑑=√64+36𝑑=√100𝑑=10
𝑟 𝑦−𝑎𝑥𝑖𝑠 (𝑥 , 𝑦 )→ (− 𝑥 , 𝑦 )
30
80 180−30−80=70
4 𝑥+2 𝑦=142 𝑦=−4 𝑥+14𝑦=−2𝑥+7
𝑚=−2𝑚∥=−2
𝑦=−2𝑥+𝑏2=−2 (2 )+𝑏2=−4+𝑏6=𝑏
𝑟 𝑦=𝑥 (𝑥 , 𝑦 )→ (𝑦 ,𝑥 )𝑟 𝑦=𝑥 (−3𝑎 ,4𝑏 )→ (4𝑏 ,−3𝑎 )
A
M
5
25
2
B
𝑦=4
XX
P
Q
R2 𝑥
3 𝑥
5 𝑥
2 𝑥+3 𝑥+5 𝑥=18010 𝑥=180𝑥=18
2 𝑥=2 ∙18=363 𝑥=3 ∙18=545 𝑥=5 ∙18=90
𝑚
𝑛
If the diagonals do not bisect each other, then the quadrilateral can not be a parallelogram!
𝑥+2 𝑦=32 𝑦=−𝑥+32 𝑦2
=−𝑥2
+32
𝑦=−12𝑥+32
𝑚=−12
𝑚⊥=2
𝑉= h𝐵𝐵=𝑏𝑎𝑠𝑒𝑎𝑟𝑒𝑎𝐵=
12h𝑏
𝐵=12∙6 ∙4
𝐵=12
𝑉=12 ∙10𝑉=120
6 2 4
𝑥
𝑥
8
𝐴𝐸 ∙𝐸𝐵=𝐶𝐸 ∙𝐷𝐸8 ∙ 4=𝑥 ∙𝑥32=𝑥2
√32=√𝑥2√32=𝑥
√16 ∙2=𝑥4√2=𝑥
𝑆=(𝑛−2 )180𝑆=(6−2 )180𝑆=(4 )180𝑆=720
h𝐸𝑎𝑐 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟=7206
h𝐸𝑎𝑐 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟=120
𝑂𝑅h𝐸𝑎𝑐 𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟=
3606
h𝐸𝑎𝑐 𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟=60
h𝐸𝑎𝑐 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟=180−60h𝐸𝑎𝑐 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟=120
𝑚𝐴𝐵=𝑦2− 𝑦1𝑥2−𝑥1
𝑚𝐴𝐵=6−20−8
𝑚𝐴𝐵=4−8
𝑚𝐴𝐵=−12
𝑚⊥=2
𝑀𝑖𝑑𝑥=𝑥2+𝑥12
𝑀𝑖𝑑𝑥=8+02
𝑀𝑖𝑑𝑥=82
𝑀𝑖𝑑𝑥=4
𝑀𝑖𝑑𝑦=𝑦2+𝑦 12
𝑀𝑖𝑑𝑥=2+62
𝑀𝑖𝑑𝑥=82
𝑀𝑖𝑑𝑥=4
𝑀𝑖𝑑𝐴𝐵=(4,4 )
𝑦=𝑚𝑥+𝑏4=2 ∙4+𝑏4=8+𝑏−4=𝑏
𝑦=2 𝑥−4
𝑎2+𝑏2=𝑐2
72+𝑥2= (𝑥+1 )2
𝑥2+49=(𝑥+1 ) (𝑥+1 )𝑥2+49=𝑥2+𝑥+𝑥+1𝑥2+49=𝑥2+2𝑥+1−𝑥2−𝑥2
49=2𝑥+148=2𝑥24=𝑥𝑥+1=24+1=25
In a trapezoid, the bases are parallel.
Parallel lines intercept congruent arcs between them.
180−80=1001002
=50
5050
𝐵𝐶=50
𝑉=43𝜋 𝑟3
𝑉=43𝜋 ∙93
𝑉=43𝜋 ∙729
𝑉=972𝜋
𝐶𝑒𝑛𝑡𝑒𝑟− (5 ,−4 )𝑟=6
(𝑥−5 )2+(𝑦+4 )2=62
(𝑥−5 )2+(𝑦+4 )2=36
Statements Reasons
1.∠𝐴𝐶𝐵≅∠𝐴𝐸𝐷 1. Given
2 .∠𝐴≅∠𝐴 2. Reflexive Postulate
3. Δ 𝐴𝐵𝐶 Δ 𝐴𝐷𝐸 3. AA AA
A
B
C
6
4
3
2 6
4
3
2 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 : (7,5 )
180−70=1101102
=55
55
55
180−55=125
125
180−125−28=27
27
is not isosceles because all the angles have different measures, which means all the sides have different lengths, making the triangle scalene, not isosceles.
𝐷2 𝑇 −3,1
G’’
S’’
H’’
𝑥+2𝑥
=𝑥+64
4 (𝑥+2 )=𝑥 (𝑥+6 )4 𝑥+8=𝑥2+6 𝑥−4 𝑥−4 𝑥
8=𝑥2+2𝑥−8−80=𝑥2+2𝑥−80=(𝑥+4 ) (𝑥−2 )𝑥+4=0𝑥=−4
𝑥−2=0𝑥=2Reject
x
y
A
B
C
10
8
5
410
4
5
2D E
F
𝑚=𝑟𝑖𝑠𝑒𝑟𝑢𝑛
𝑚𝐴𝐷=54
𝑚𝐷𝐸=06=0
𝑚𝐹𝐸=54
𝑚𝐴𝐹=06=0
Since the opposite sides of ADEF have equal slopes, they are parallel. Since the opposites sides of ADEF are parallel, ADEF is a parallelogram.
𝑑𝐴𝐷 :𝑎2+𝑏2=𝑐2
𝑑𝐴𝐷 :52+42=𝑐2
25+16=𝑐241=𝑐2
√ 41=𝑐6.4=𝑐𝑑𝐷𝐸=6
Since two consecutive sides of parallelogram ADEF are not congruent, then ADEF is not a rhombus.
6.4
6