MEE3 Unit 10 Terms and Practice/review PDF

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1

MEE3 Unit 10 Terms and Practice Formulas Circumference = distance around a circle 𝑪 = 𝟐𝝅𝒓, where C if the circumference and r is the radius. * Note that 2𝜋𝑟 = 𝜋𝑑 , where d is the diameter, because 𝑑 = 2𝑟 (i.e., the diameter is twice the radius or the radius is ½ the diameter)

r

𝒓=

𝑪

find the radius given the circumference

𝟐𝝅

Arclength of a circle = s 𝒑𝒂𝒓𝒕

𝒂𝒓𝒄 𝒍𝒆𝒏𝒈𝒕𝒉

𝒘𝒉𝒐𝒍𝒆

𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆

𝒔 𝟐𝝅𝒓

=

=

𝒂𝒏𝒈𝒍𝒆° 𝟑𝟔𝟎°

𝒂𝒏𝒈𝒍𝒆° 𝟑𝟔𝟎°

Equation of a circle: (𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐 where (h,k) is the center and r is the radius Area = space inside of a figure Circle

Triangle

Parallelogram (and Rectangle)

Trapezoid a

r h b 𝑨 = 𝝅𝒓

𝟐

𝟏 𝑨 = 𝒃𝒉 𝟐 Area of a Semicircle

𝑨=

h

h

𝟏 𝟐

𝝅𝒓𝟐

𝑨 = 𝒃𝒉 Area of a Sector

𝒑𝒂𝒓𝒕

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓

𝒘𝒉𝒐𝒍𝒆

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒄𝒊𝒓𝒄𝒍𝒆

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓

Also,

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒄𝒊𝒓𝒄𝒍𝒆

b 𝟏 𝑨 = 𝒉(𝒂 + 𝒃) 𝟐

b

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒄𝒊𝒓𝒄𝒍𝒆 𝒂𝒓𝒄 𝒍𝒆𝒏𝒈𝒕𝒉

=

= 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒓

Pythagorean Theorem: c2 = a2 + b2 Distance Formula: 𝒅 = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐

=

𝒂𝒏𝒈𝒍𝒆° 𝟑𝟔𝟎°

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓 𝝅𝒓𝟐 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓 𝝅𝒓𝟐

= =

𝒂𝒏𝒈𝒍𝒆° 𝟑𝟔𝟎° 𝒂𝒓𝒄 𝒍𝒆𝒏𝒈𝒕𝒉 𝟐𝝅𝒓

2

Area of composite figures. Composite figures are composed of two or more shapes. You can add to find area or you can subtract to find area. Ex. Find the area of the following figure. There are two shapes – a triangle on top and a rectangle on the bottom. Find the area of each and add them together to find the total area. Triangle at the top has base 11 and height 4. 𝟏 𝟏 𝑨 = 𝟐bh = 𝟐 ∙ 𝟏𝟏 ∙ 𝟒 = 𝟐𝟐 Rectangle at bottom has length 11 and width 9. A = lw = 11∙ 9 = 99 Area of figure = 22 + 99 = 121 in2 Area has square units.

Ex. Find the area of the following figure. There are two shapes – half a circle over a rectangle Find the area of each and add them together to find the total area. Half circle has diameter of 7 – 3 = 4. Radius = 2. 𝟏 𝟏 A = 𝟐 𝝅𝒓𝟐 = ∙ 𝟑. 𝟏𝟒 ∙ 𝟐𝟐 = 𝟔. 𝟐𝟖 𝟐 Rectangle at bottom has length 7 and width 2. A = lw = 7∙2 = 14 Area of figure = 6.28 +14 = 20.28 cm2 Area has square units.

Ex. Find the area of the following figure. There are three shapes – two half circles = 1 circle and a square Find the area of each and add them together to find the total area. Half circles have diameter of 2. Radius = 1. A = 𝝅𝒓𝟐 = 𝟑. 𝟏𝟒 ∙ 𝟏𝟐 = 𝟑. 𝟏𝟒 Square has length 2 and width 2. A = lw = 2∙2 = 4 Area of figure = 3.14 + 4 = 7.14 in2 Area has square units.

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Ex. A rectangular piece of paper 10cm by 15cm has a circle cut out of it whose diameter is 8 cm. What is the area of rectangle without the circle? There are two shapes – a circle inside a rectangle. Find the area of each and subtract to find the area. Rectangle has length 10 and width 15. A = lw = 10 ∙ 𝟏𝟓 = 150 Circle has diameter of 8. Radius =4. A = 𝝅𝒓𝟐 = 𝟑. 𝟏𝟒 ∙ 𝟒𝟐 = 𝟓𝟎. 𝟐𝟒 Area of figure = 150 – 50.24 = 99.76 cm2 Area has square units. Ex. Find the area of the shaded region. There are two shapes – a triangle inside a rectangle. Find the area of each and subtract to find the shaded area. Rectangle has length 8 ft and width 5 ft. A = lw = 8 ∙ 𝟓 = 40 Triangle has base 36/12 = 3 ft and height is 27/12 = 2.25 ft (Use the same units for both figures either feet or inches.) 𝟏 𝟏 𝑨 = 𝟐bh = ∙ 𝟑 ∙ 𝟐. 𝟐𝟓 = 𝟑. 𝟑𝟕𝟓 𝟐 Area of figure = 40 – 3.375 = 36.625 ft2 Area has square units. Ex. Find the area of the shaded region. There are two shapes – a circle inside another circle. Find the area of each and subtract to find the shaded area. Outside circle has radius = 3 + 2 = 5. A = 𝝅𝒓𝟐 = 𝟑. 𝟏𝟒 ∙ 𝟓𝟐 = 𝟕𝟖. 𝟓 Inside circle has radius = 3. A = 𝝅𝒓𝟐 = 𝟑. 𝟏𝟒 ∙ 𝟑𝟐 = 𝟐𝟖. 𝟐𝟔 Area of figure = 78.5 – 28.26 = 50.24 cm2 Area has square units. Area of a sector. Ex. Find the area of the sector of a circle with angle 100 degrees and radius 7 cm. 𝒑𝒂𝒓𝒕

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓

𝒘𝒉𝒐𝒍𝒆

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒄𝒊𝒓𝒄𝒍𝒆 𝒙 𝝅𝒓𝟐 𝒙 𝟒𝟗𝝅

=

=

𝒂𝒏𝒈𝒍𝒆° 𝟑𝟔𝟎°

𝟏𝟎𝟎

𝒙

𝟑𝟔𝟎

𝝅∙𝟕𝟐

=

𝟏𝟎𝟎 𝟑𝟔𝟎

𝟏𝟎𝟎 𝟑𝟔𝟎

360x = 4900𝜋 𝑥=

=

4900𝜋 360

=

Cross multiply. 245 18

𝜋 = 42.7 cm2

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Ex. Find the area of the sector of a circle with arc length 4.2 ft and radius 4 ft. 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟 𝑥

=

𝑥 16𝜋

=

4.2

Use part to whole proportion

𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒

=

𝜋𝑟 2 𝜋∙4 2

𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ

=

𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 2𝜋𝑟

x = area of sector, 4.2 = arc length, r = 4

2𝜋∙4 4.2

cross multiply and solve

8𝜋

8𝜋𝑥 = 67.2𝜋 𝑥=

67.2𝜋 8𝜋

= 8.4 𝑓𝑡

Ex. Find the radius of the circle if the arc length is 15 cm and the angle is 27 degrees.. 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 15 2𝜋𝑟

=

27

=

𝑛° 360°

arc length = 15, n = 27

360

54𝜋𝑟 = 5400 cross multiply and solve 𝑟=

5400 54𝜋

= 31.8 𝑐𝑚

Ex. Find the radius of the circle if the arc length is 24π and the angle is 120 degrees. 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑛° = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 360° 24𝜋 2𝜋𝑟

=

120 360

arc length = 24π, n = 120

240𝜋𝑟 = 8640𝜋 𝑟=

Pythagorean Theorem – the square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs.

8640𝜋 240𝜋

= 36

cross multiply and solve

5

Ex. Find c.

Ex. Find a.

𝑎 = √𝑐 2 − 𝑏2 𝑎 = √8.52 − 4.22 𝑎 = 7.39

Ex. Find h. ℎ = √102 − 52 h= √100 − 25 ℎ = 8.7 𝑚

Ex. Find the length of the diagonal of the square.

𝑐 = √𝑎2 + 𝑏 2 c= √82 + 82 𝑐 = 11.31

Ex. Find the radius and center of the following. A. (𝑥 − 4)2 + (𝑦 + 2)2 = 16 Radius is 4 and center is (4, - 2) B. 𝑥 2 + 𝑦 2 = 15 Radius is √15 and center is (0, 0) Ex. Write the equation of the circles. A. Radius is √35 and center is (- 5,7) (𝑥 + 5)2 + (𝑦 − 7)2 = 35 B. Radius is 4 and center is (- 3, - 4) (𝑥 + 3)2 + (𝑦 + 4)2 = 16

6

Ex. Find the distance between (-2, 4) and (-3, -1)

Practice 1. Find the area of the following figure. A. 40

B. 36

C. 34.5

D. 35.5

2. Find the area of the shaded region of the figure.

A. 10.4

B. 17.4

C. 45.4

D. 28.8

A. 1579 ft2

B. 3463 ft2

C. 1745 ft2

D. 1991 ft2

3. Find the area of the figure.

7

4. Find the area of the figure.

A. 36 in2

B. 33 in2

C. 45 in2

D. 24 in2

5. Find the area of the shaded region of the figure. A. 265

B. 17.6

C. 29.5

D. 8.5

A. 31.4

B. 600

C. 10

D. 5.2

A. 3.9

B. 15.7

C. 98.6

D. 31.4

A. 53.1 cm

B. 675 cm

C. 26.5 cm

D. 1.9 cm

6. Find the area of the sector.

7. Find the area of the sector.

8. Find the radius of the figure.

25 cm

8

9. Find h. A. 61 ft

B. 875 ft

C. 36 ft

D. 15 ft

10. Find the length of the diagonal of the tablet.

A. 3.6 in

B. 13 in

C. 1 in

D. 9.2 in

Answers: 1-B, 2-A, 3-D, 4-A, 5-C, 6-A, 7-B, 8-A, 9-C, 10-D...