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 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 C d (Remember d 2r ) 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. 0.328m 80 cm mm 3.1 Calculating Areas Rectangle Parallelogram Square Triangle b2 b1 r Trapezoid Circle 3.1 Area - Square Units 4. What is the area of a circular region whose diameter is 6 cm? Formula: A = r 2 (3) 2 A. 36 sq. cm C. 12 sq. cm If d = 6, then r = 3 B. 6 sq. cm D. 9 sq. cm 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=2LH +2WH +2LW 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=2LH +2WH +2LW 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 Rectangular Solid Cylinder Cone r r Sphere V=lwh V r h 1 2 V r h 2 3 4 3 V r 3 3.1 Example of Volume 8. What is the volume of a sphere with a 12 inch diameter? 4 If d = 12, 3 Formula: V r then r = 6 3 4 3 Since (6)(6)(6)= 216, the V (6) 3 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 3D 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 c a b 2 2 2 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? c a b 2 2 2 c 2 (12)2 (5)2 144 25 169 c 13 and 3 wires x 13 ft = 39 ft 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 = 1/2xbh, then 6 x 1/2 x bh = 3 bh A. 3h+b B. 6(h+b) C. 6hb h b D. 3hb 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 55 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 70 C 3.3 Angle Measures 1. B S Theorem All B’s are = , All S’s are = B + S = 180 B S S B B S S B 2. Perpendicular lines intersect to form right angles. 3.3 Angle Measures 1 2 L1 The parallel lines are 3 4 cut by transversal T 5 6 L2 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 L1 The parallel lines are 3 4 cut by transversal T 5 6 L2 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 L1 and L2 are parallel? After using the BS theorem, angle T does = 75 and angle S=105 60 75 L1 45 R 75 60 45 105 S T7545 135 L2 75 V 105 135 45 A.Since mT 75,mS 60 B.Since mT 75,mS 105 C.mV mR D.None 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. mA = mE 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, ii, 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. mA = mE 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, ii, 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. mA = mE 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, ii, iii REMEMBER MATH IS FUN AND … YOU CAN DO IT