GEOMETRY CHAPTER 3 Geometry Measurement 3 1 Measuring

GEOMETRY CHAPTER 3

Geometry & Measurement 3. 1 Measuring Distance, Area and Volume 3. 2 Applications and Problem Solving 3. 3 Lines, Angles and Triangles

3. 1 Rounding Measurements To round: 1. Underline the place 2. If number to the right of the underlined place is 5 or more, add one 3. Otherwise, do not change 4. Change all digits to the right of underlined number to zeros

3. 1 Rounding Example: Round 38. 67 centimeters to the nearest centimeter 1. 38. 67 2. First number to the right of 8 is “ 6”, so add one to 8 4. Change all digits to the right to 0’s. The answer is, 39. 00 or 39

3. 1 Calculating Distances Linear Measure - a distance which could be around a polygon (perimeter) or around a circle (circumference) Perimeter - sum of the lengths of the sides Circumference - distance around circle

3. 1 Metric Measures Measure can be in U. S. system (yd, ft, etc. ) or metric (cm, m, etc) King km Henry hm Died dam kilometer hectometer dekameter Drinking Chocolate dm cm decimeter centimeter Monday m meter Milk mm millimeter

3. 1 Metric Measures 1 km = 1000 m 1 dm = 0. 1 m 1 hm = 100 m 1 cm = 0. 01 m 1 dam = 10 m 1 mm = 0. 001 m

3. 1 Linear Distance 2. What is the distance around the polygon, in meters? 78 cm 78 + 95 + 80 + 75 = 328 cm km hm dam m A. 328 m C. 3. 28 m 75 cm 95 cm dm cm B. 32. 8 m D. 0. 328 m 80 cm mm

3. 1 Calculating Areas Rectangle Parallelogram Square Triangle Trapezoid r Circle

3. 1 Area - Square Units 4. What is the area of a circular region whose diameter is 6 cm? Formula: If d = 6, then r = 3

3. 1 Examples of Area Surface area of a rectangular solid H There are 6 faces of the solid L W Front/back Sides (Left/Right) Top/Bottom A=2 LH +2 WH +2 LW Square units

3. 1 Examples of Area 6. What is the surface area of a rectangular solid that is 12 in. long, 5 in. wide and 6 in. high? H=6 L=12 W=5 A=2 LH +2 WH +2 LW A=2(12)(6)+2(5)(6)+2(12)(5) A. 360 cubic in. B. 324 sq. in. C. 324 cubic in. D. 360 sq. in.

3. 1 Volume - Cubic Units h h w l r h r Rectangular Solid Cylinder Cone r Sphere

3. 1 Example of Volume 8. What is the volume of a sphere with a 12 inch diameter? If d = 12, Formula: then r = 6 Since (6)(6)(6)= 216, the only reasonable ans. is C

3. 1 Identifying the Unit 9. Which of the following would not be used to measure the amount of water needed to fill a swimming pool? A. Cubic feet B. Liters C. Gallons D. Meters linear Think of “volume” as capacity or filling up the inside of a 3 D figure.

3. 2 Application Example 1. What will be the cost of tiling a room measuring 12 ft. by 15 ft. if square tiles cost $2 each & measure 12 in. ? Since 12 inches = 1 ft, one tile is 1 ft on each side or 1 sq. ft. Area room: A = bh; (12)(15) = 180 sq ft And (180)($2) = $360 cost A. $180 B. $4320 C. $360 D. $3600

3. 2 Pythagorean Theorem For any RIGHT TRIANGLE a c b Side opposite the right angle is the hypotenuse “c”

3. 2 Pythagorean Theorem 3. A TV antenna 12 ft. high is to be anchored by 3 wires each attached to the top of 12 antenna and to pts on the roof 5 ft. from base of the antenna. If wire costs $. 75 per ft, what will be the cost? Cost is. 75 x 39 =$29. 25 c 5 A. $27. 00 B. $29. 25 C. $9. 75 D. $38. 25

3. 2 Infer & Select Formulas 7. The figure shows a regular hexagon Select the formula for total area Total area is the area of the 6 identical triangles. If area of 1 triangle = 1/2 xbh, then 6 x 1/2 x bh = 3 bh A. 3 h+b B. 6(h+b) C. 6 hb h b D. 3 hb

3. 3 Lines; Angles; Triangles ANGLES straight angle 180 right angle 90 obtuse > 90, < 180 acute angle < 90 comp. sum to 90 supp. sum to 180 vertical angles-equal TRIANGLES Right triangle Acute triangle Obtuse triangle Scalene triangle Isosceles Equilateral

3. 3 Properties Example 2. What type of triangle is ABC? Since sum of angles of triangle = 180, and 55 + 70 = 125, then angle C = 180 - 125 = 55. If 2 angles = , then isosceles. A. Isosceles C. Equilateral B. Right D. Scalene C

3. 3 Angle Measures 1. B S Theorem All B’s are = , All S’s are = B + S = 180 B S S B 2. Perpendicular lines intersect to form right angles.

3. 3 Angle Measures 1 2 L 1 The parallel lines are 3 4 cut by transversal T 5 6 L 2 7 8 Corresponding angles are = T Terminology 1 and 5, 3 and 7, 2 and 6, 4 and 8 Vertical angles are = 1 and 4, 3 and 2, 6 and 7, 5 and 8

3. 3 Angle Measures 1 2 L 1 The parallel lines are 3 4 cut by transversal T 5 6 L 2 7 8 T Alternate interior angles are = Terminology 4 and 5, 3 and 6 Alternate exterior angles are = 1 and 8, 2 and 7

3. 3 Angle Measures 3. If 2 angles of a triangle are = , then sides opposite are = 4. If 2 sides of a triangle are =, then angles opposite are =

3. 3 Examples 6. Which statement is true for the figure shown at the right given that L 1 and L 2 are parallel? After using the BS theorem, angle T does = 75 and angle S=105 60 75 L 1 45 R 75 60 45 105 S T 7545 135 L 2 75 V 105 135 45

3. 3 Similar Triangles Two triangles are similar if all angles are = and sides proportional A 10. Which statements are true? 7. 5 x i. m A = m E 40 D ii. AC = 6 40 B C iii. CE/CA = CB/CD 5 4 E Since m D=m B and DCE and ACB are Vertical angles m A=m E A. i only B. ii only C. i and ii only D. i, iii

3. 3 Similar Triangles Two triangles are similar if all angles are = and sides proportional A 10. Which statements are true? 7. 5 x i. m A = m E 40 D ii. AC = 6 40 B C iii. CE/CA = CB/CD 5 4 E The triangle are similar, thus ratios of corresponding sides are =. x/4 = 7. 5/5 thus x= 4(7. 5)/5 = 6 A. i only B. ii only C. i and ii only D. i, iii

3. 3 Similar Triangles Two triangles are similar if all angles are = and sides proportional A 10. Which statements are true? 7. 5 x i. m A = m E 40 D ii. AC = 6 40 B C iii. CE/CA = CB/CD 5 4 E The triangle are similar, thus ratios of corresponding sides are =. CE/CA = CD/CB thus iii is false! A. i only B. ii only C. i and ii only D. i, iii

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