NOTES: A quick overview of 2-D geometry

Transcription

1 NOTES: A quick overview of 2-D geometry Wat is 2-D geometry? Also called plane geometry, it s te geometry tat deals wit two dimensional sapes flat tings tat ave lengt and widt, suc as a piece of paper. You sould e familiar wit te following 2-d sapes: rectangle square parallelogram rigt trapezoid pentagon, exagon, octagon circle equilateral scalene d isosceles 1 2 r 2 sets of congruent, parallel sides 4 rigt angles actually just a special kind of parallelogram one tat as 4 rigt angles 2 sets of parallel sides 4 congruent sides & 4 rigt angles actually just a special kind of rectangle one tat as equal lengt & widt 2 sets of congruent, parallel sides eigt () is straigt down from te igest point to form a rigt angle wit te ase; never te slanty side alf a parallelogram/rectangle/square eigt () is straigt down from te igest point to form a rigt angle wit te ase; never te slanty side Triangles are classified y sides: equilateral (all sides equal), isosceles (2 sides equal), and scalene (no sides equal). Tey re also classified y angle: rigt (one 90 ), otuse (one >90 ), acute (all <90 ) 2 ases: 1 & 2 eigt () is straigt down from 1 to 2; never te slanty side alf a parallelogram wit ase sides, 6 sides, 8 sides diameter (d) = distance from 1 side to te oter troug te center radius (r) = ½ te diameter; distance from te center to te edge D. Stark /20/2017 OVERVEIW: 2-D Geometry 1

2 Wat are some useful its of vocaulary to know? polygon: a closed 2-d sape made wit straigt line segments regular polygon: a polygon wit all sides te same lengt regular exagon exagon tat isn t regular ase (lowercase ): te side of certain 2-d sapes at rigt angles to (perpendicular to) te eigt; usually ut not always te ottom congruent: identical in sape and size; te word we use for equal wen we re talking aout sapes instead of numers; symolized y ( ABC DEF) and as marks on diagrams A D B C E F parallel: always te same distance apart; symolized y (L1 L2) L1 L2 perpendicular: at rigt angles (90 ); symolized y (L3 L4) and on diagrams wit L3 L4 Wat is perimeter, and ow do you calculate it? Perimeter is te distance around a sape. Tink of counting your steps wit a pedometer as you walk around. Perimeter is always measured in regular units: in., ft., cm, m. It s a distance. EXAMPLE #1: Wat s te perimeter of te rectangle elow? m You could use te formula from te formula seet: P = 2L + 2w = 2(6.3) + 2(2.1) = 16.8 m Or you could just add up all te sides. D. Stark /20/2017 OVERVEIW: 2-D Geometry 2

3 EXAMPLE #2: Wat s te perimeter of a regular octagon wit a side lengt of 7 in? Tere s no formula ere. Since an octagon as 8 sides, te perimeter is 8(7) = 56 in. EXAMPLE #3: Wat s te perimeter of te room sown elow? Rememer to find te unmarked sides! Ten add tem up: P = = 84 ft. EXAMPLE #4: If te perimeter of a rectangle is 12.6 m and te lengt is 4.2 m, wat s te widt? 4.2 m Witout algera: = 8.4 Since P = 12.6, tere s = 4.2 left to e divided etween te two? sides. Te widt is 2.1 m Wit algera: P = 2L + 2w 12.6 = 2(4.2) + 2w 12.6 = w?? 4.2 m [from te formula seet] 4.2 = 2w w = 2.1 m D. Stark /20/2017 OVERVEIW: 2-D Geometry 3

4 Wat is area, and ow do you calculate it? Area is te space inside a 2-d sape. Tink of covering someting wit little 1 ft 1 ft area rugs. Area is always measured in square units: in 2, ft 2, cm 2, m 2. Don t confuse perimeter and area! See te formula seet. EXAMPLE #1: Wat s te area of te rectangle elow? m A = Lw = (6.3)(2.1) = EXAMPLE #2: Wat s te area of a square wit side lengt 11 in? You could use te formula seet: A = s 2 = 11 2 = 121 in 2 Or you could recognize tat a square is just a special kind of rectangle, namely, one wit lengt te same as widt. A = Lw = (11)(11) = 121 in 2 EXAMPLE #3: Wat s te area of te parallelogram elow? 2.1 m 6. First, notice tat if you slide te triangular piece to te rigt, you get te rectangle of EXAMPLE #1 so te area sould e te same as it was tere. To get te area ere, you multiply (6.3)(2.1) = Instead of lengt and widt we ave ere ase and eigt. Base and eigt always form a rigt angle. Heigt is never te slanty side. Note tat a rectangle is actually just a special kind of parallelogram, namely, one in wic tere are 4 rigt angles. You can tink of te area of a rectangle as really A =, too. D. Stark /20/2017 OVERVEIW: 2-D Geometry 4

5 6 ft 4 m EXAMPLE #4: Wat s te area of te on te rigt? First, notice tat a is really alf a parallelogram. Since te area of a parallelogram is A =, te area of a sould e alf tat, and it is: A = ½ Just as for parallelograms, eigt on s is always perpendicular to te ase. It s never te slanty side. For te original, te ase is 6 m (3m + ) and te eigt is 4 m. A = ½ = ½ (6)(4) = 12 m 2 Rememer tat taking ½ of someting (multiplying it y ½) is te same as dividing y 2. So you could just multiply ase y eigt and divide te result y 2 instead: (6)(4) = 12 m 2 2 EXAMPLE #5: Wat s te area of te trapezoid on te rigt? We can t just do A = wit te ottom ase since tat would e te area of te ig dotted rectangle elow, and tat s too ig. But if we used te oter ase (perpendicular to te eigt), we d get a rectangle tat s too small. 4 m 8 ft 6 ft 16 ft 8 ft 16 ft 6 ft 16 ft 8 ft Wat we need is te average of te ig ase and te little ase. Well, to find te average of 2 numers, you add tem and divide y 2. Anoter way to say tat is to add tem and take ½ of te result. From tis we can make sense of te formula for te area of a trapezoid: A = ½ (1 + 2) [I ve moved te to make te formula more intuitive.] = ½ (16 + 8)(6) = 72 m 2 D. Stark /20/2017 OVERVEIW: 2-D Geometry 5

6 Tis page as arder ackwards examples. EXAMPLE #6: If te area of a rectangle is cm 2 and te widt is 3.4 cm, wat s te lengt? A = Lw = 3.4L L = 4.8 m EXAMPLE #7: If te area of a square is 4 25 mi, wat s te lengt of 1 side? A = s = s2 s = 4 25 = 2 5 mi. EXAMPLE #8: If te area of te parallelogram sown is 0.08 m 2, wat s te ase? A = 0.08 = 0.2 = 0.4 m EXAMPLE #8: If a rigt as one leg lengt of 12 inces and an area of 30 in 2, wat s te lengt of te oter leg? Note tat since it s a rigt, one leg is te ase and te oter is te eigt. A = ½ 30 = ½ (12) = 5 in EXAMPLE #8: If te area of a trapezoid is 36 yds 2, its eigt is 4 yds, and one ase is 10 yds, wat s te oter ase? A = ½ (1 + 2) 36 = ½ (10 + 2) (4) 36 = 2 (10 + 2) 36 = = 22 2 = 8 yds. 0.2 m 12 in D. Stark /20/2017 OVERVEIW: 2-D Geometry 6

7 How do you find te circumference & area of circles? Calculations wit circles all involve te irrational numer, a decimal tat goes on FOREVER witout repeating. Te GED test formula seet and many ooks suggest using 3.14 as a good enoug approximation of. Circumference for circles is like perimeter for polygons. Imagine putting a fence around a circle and finding its lengt. Circumference is a lengt, so it s in regular units: in., ft., m, etc. Tere are 2 formulas on your seet for circumference: 2 r and d Since te diameter (d) is twice te radius (r), tese are equivalent. Use wicever is more convenient. EXAMPLES: 1) Wat s te circumference of a circle wit a radius of 5 m? C = 2 r = 2(3.14)(5) = 31.4 m 2) Rounded to te nearest foot, wat s te circumference of a circle wit a diameter of 3 ½ ft? C = d = (3.14)(3 ½) = ft 3) If a circle as a circumference of cm, wat s te radius? C = 2 r = 2(3.14)(r) r = 5.6 cm Te circle area formula on your seet (A = r 2 ) is only in terms of radius. If you ve got te diameter, rememer to divide tis y 2 to get te radius. Rememer tat area is ALWAYS in square units. EXAMPLES: 1) To te nearest square kilometer, wat s te area of a circle wit a diameter of 5.5 km? A = r 2 = (3.14)(2.75) 2 = km 2 2) Wat s te diameter of a circle wit an area of cm 2? A = r = (3.14)r 2 r 2 = = 16 r = 16 = 4 cm D. Stark /20/2017 OVERVEIW: 2-D Geometry 7

8 How do you find te area of composite sapes? Composite (or compound) sapes are simply several sapes put togeter. To find te area of tese, just add or sutract pieces wicever is appropriate or most intuitive for you. EXAMPLE: Find te area of te room sown elow. Addition metod Sutraction metod 5 ft 10 ft area room = area ig + area little = (24)(13) + (10)(5) = 362 ft 2 10 ft area = area wole area left area rigt room = (24)(18) (10)(5) (5)(4) = 362 ft 2 How do you use te Pytagorean Teorem? Te Pytagorean Teorem states a relationsip among te side lengts of a rigt. It s on your formula seet. a = c 2 c a a & are te lengts of te legs of a rigt [It doesn t matter wic one you call a and wic you call.] c is te lengt of te ypotenuse (te longest side, wic is opposite te rigt angle) [It does matter tat you call te longest side c.] Geometrically, te teorem says tat te area of te square on one leg plus te area of te square on te oter leg add up to te area of te square on te ypotenuse. Mostly, toug, you ll e tinking of squares algeraically. D. Stark /20/2017 OVERVEIW: 2-D Geometry 8

9 EXAMPLE #1: Wat s te missing side lengt in te figure elow? [Find te ypotenuse.] 3 in.? 4 in. a = c = c = c 2 25 = c 2 c = 25 = 5 in. Most Pytagorean prolems don t work out so nicely wit all wole numers. Te ones tat do ave a special name Pytagorean triples. It s wort memorizing te most common one: ecause lots of questions use tese numers or multiples of tem (like douling everyting for or tripling everyting for ). If you can spot a Pytagorean Triple, you can save yourself time calculating. o oter Pytagorean Triples: ; ; EXAMPLE #2: Wat s te missing side lengt in te figure elow? [Find te leg.] 5 in.? 13 in. a = c = = = 144 = 144 = 12 in. Finding a leg is arder tan finding te ypotenuse since you need to solve a one-step equation to isolate 2 and ten. EXAMPLE #3: Wat s te ypotenuse of a rigt wit a eigt of 4.2 cm and a ase of 6.8 cm? Round your answer to te nearest tent. a = c 2 (4.2) 2 + (6.4) 2 = c = c 2 c = 58.6 = etc. 7.6 cm. GED is a registered trademark of te American Council on Education (ACE) and administered exclusively y GED Testing Service LLC under license. Tis material is not endorsed or approved y ACE or GED Testing Service. D. Stark /20/2017 OVERVEIW: 2-D Geometry 9