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