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Y M IN CH Y R LD A FREN T E R LIZ M O E O WG Block 3 Geometry H, T3 & T4 2012
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ExplanationGeometry is not a concept only used in math class, itsimply helps us identify real world applications ofthings like shapes and lines. We can use the lessonswe learn in class to discover new things about theworld around us. For example, a stop sign is aoctagon, a regular polygon. The legs of tables andchairs will often be parallel lines. This PowerPoint willshow examples of Geometry I found in my life. I willexplain how each picture is an example of Geometryand where I found it. 2
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Decorative Plate – Circle (L. 23)This is a picture of adecorative plate that hangsin my house.The plate is an example of acircle. The center point(Point A) of the circle wasused in the design by theartist to determine where topaint the birds. Definitions: A center of a circle is the point inside a circle that is equidistant from every point on the circle. A circle is a closed plane curve consisting of all points at a given distance from a point within it called the center. 3
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Table Lamp – Decagon (L. 15)This is a stain glass tablelamp that sits next to thereading chair upstairs in mycomputer room.This lamp is a regulardecagon. On the lamp, thecongruency of the sides isshown with pink congruencymarkings, the diagonals,forming central angles aremarked with blue, and thesides are yellow. Definitions: A decagon is a ten-sided polygon. A regular polygon is a polygon that is both equilateral (all the sides are the same length) and equiangular (all the 4 angles are the same measure).
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Stove Fan – Frustum of a Pyramid (L.103)This picture is of the fanabove my stove, which is usedto bring better air circulationwhen cooking food thatproduces a lot of smoke.The red lines represent theshape of the fan, whichcreates a frustum. The bluelines signify where the top ofthe pyramid would be, if itwere complete. Definitions: A frustum of pyramid is a part of a pyramid with two square parallel bases.In t e r e s t in g N o t e * A pyramid is a polyhedron formed by a polygonal base andV o lu m e o f a F r u s t u m : 5 triangular lateral faces that meet at a common vertex.V = 1/3 h ( B 1 + √ B 1( B 2 ) +
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Back Yard Fence – Perimeter (L. 19)At my house, there is a squareback yard. Each side of the yardis 40 ft. Around that yard, Thereis a wooden fence, which isshown in the picture here.However, there is a 4 footpathwayThe fence represents theperimeter of the yard, which in asquare can be found with theformula “P = 4s” where s is thelength of the sides, in the caseWorking out40 ft.of my yard, the Formula: Definitions:Step 1: P = 4s – 4 A perimeter is the sum of the side lengths of a closedStep 2: P = 4(40) – 4 plane figure.Step 3: P = 160 – 4 6Step 4: P = 156 ft
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Coat Hanger – Rhombus (L.52)This is a coat hanger that Iuse daily in my room to hangclothes on.When pulled apart, the coathanger forms a rhombus.The diagonals of therhombus bisect each other,forming congruent segmentsand right angles. Definitions: A rhombus is a quadrilateral with four congruent sides. The Properties of a Rhombus state that the diagonals of a rhombus are perpendicular and that each diagonal of a rhombus bisects opposite angles. Because opposite angles of a rhombus are equal, 7 when they are bisected by a diagonal, four congruent angles result.
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CD Case – Tangent (L. 43)This is a picture of a blank CDwaiting to be used on mycomputer desk.The edge of the CD case, LineSegment AB which isrepresented in yellow, liestangent to Circle P,represented in red around theCD. The point of tangency inthis picture is Point C. Definitions: The tangent of a circle is a line in the same plane as the circle that intersects the circle at exactly one point. The point of tangency is the single point that the tangent line intersects a circle with. 8
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Nesting Boxes – Ratios (L. 41)This is a picture of somenesting boxes that Ireceived as a gift.The boxes are similarshapes because the sidesare equivalent ratios. Thesquare of the side lengthsof the square top to theheight is ¾.For instance, a proportionof these numbers wouldbe: 6/8 = 4.5/6 = 3/4 =1.5/2. Definitions: A ratio is a comparison of two quantities by division. Similar polygons are polygons whose corresponding angles are congruent and whose corresponding sides are proportional. A proportion is a statement that two ratios are equal. 9
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Window – Plane Intersections (L. 4)This is a window on the secondstory of my house, looking outinto my front yard.In this picture, there are 8planes. 3 are highlighted asplane M, plane R, and plane P.Plane M and Plane P intersectat line segment AB. Planes M,R, and P intersect at a point,Point A. Postulate 7: If two planes intersect, then their intersection is a line. Definitions: A plane is an undefined term in geometry; a flat surface that has no thickness and extends infinitely. 10
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Mirror and Door – Chord (L.43)This is a picture of a mirror inmy house that is reflecting acloset door.The seam of the closet door,represented by line segmentAB, is a chord to the circle ofthe mirror, represented asCircle P because points A andB lie on the circle. Definitions: A chord is a segment whose endpoints lie on a circle. 11
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Porch Railing – Parallel and Perpendicular Lines (L. )This is a picture of thewooden railing on my porch.The poles represent linesegments AE, BF, CG, and DH.These line segments areparallel, as signified by thewhite markings. The whiteright angles show that linesegment AD is perpendicularto line segments AE, BF, CG,and DH. 12
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Rain Gauge – Rectangular Prism (L. 59) This is a (broken) rain gauge that I have placed outside to measure the amount of rainfall. The rain gauge is in the shape of a rectangular prism. It has a two bases, one at the top and one at the bottom, and four equivalent sides. To find the volume of this prism, you would use the formula V = Bh, where B is the area of one base and h is the volume. To find the surface area, you would use the formula SA = ph + 2B, where p is theWorking out Volume: of Workingbase, h Area: the Definitions: and B perimeter the out Surface is height, is the area of one base.Step 1: V = Bh Step 1: SA = ph + 2B A rectangular prism is a prism with six rectangular faces.Step 2: V = 1(1.5)(7) Step 2: SA = 2(1 + 1.5)(7) + 2(1) (1.5) Volume is the number of non-overlapping unit cubes of a givenStep 3: V = 10.5 in ² size that will exactly fill the interior of a three-dimensional figure. Step 3: SA = 2(2.5)(7) + 2(3) Step 4: SA = 41 in Surface Area is the total area of all faces and curved surfaces of a three-dimensional figure. 13
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Macaroni Box – Nets (Investigation 5)This is a deconstructed macaronibox, found in my recycling and thebox in its completed form, foundin the pantry.The deconstructed box is a net ofthe original box. The single planedeconstruction could be folded toform the original box. The yellowsections of the net correspond tothe yellow sections of the box, thered to the red, and the blue to theblue. Definitions: A net is a diagram of the faces of a three-dimensional figure, arranged so that thee diagram can be folded to form the three- dimensional figure. 14
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Scented Candle – Cylinder (L. 62) This is a scented candle that my mother keeps on the kitchen counter. The candle is in the shape of cylinder. The height of the candle is 15 cm, and the radius, or distance from the center point of the circle base, which is the wick, to the edge of the candle is 5 cm. To find the volume of the cylinder, you use the formula V = πr²h. To find the Surface Area, the formula SA = 2πrh + 2B is used. The variable h represents the height, B represents the oareag of uthe base, andW o r k in g o u t W r k in o t Definition:Vo lu m e : Surfa c e Ar e a : r represents the radius.S t e p 1: V = π ² h r S t e p 1: S A = 2 π h + r A cylinder is a three-dimensional figure with two parallel 2 BS t e p 2 : V = π ² ( 15 ) 5 congruent circular bases and a curved lateral surface Step 2 : SA = that connects the bases.Step 3 : V = π2 5 ) ( 2 π ( 1 5 ) + 2 π5² 5( 15 ) 15 S t e p 3 : S A = 15 0 π +Step 4 : V = π3 7 5 ) ( 2 π25
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Stairs – Kite (L. 19, L. 69) This is a segment of my stairway, enabling the stairs to curve around the wall. The shape of the step outlined in yellow is a kite. The dotted lines represent the hidden parts of the shape, as it was impossible to capture the entire shape from any angle. Two pairs of the kite are congruent and its diagonals form four right angles.P r o p e r t ie s o fK it e s : Definitions: A kite is a quadrilateral with exactly two pairs ofT h e d ia g o n a ls congruent consecutive sides. 16o f a k it e a r ep e r p e n d ic u la r .
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Salt Shaker – Regular Polygons (L. 15 )This is an aerial view of asalt shaker that I found onmy kitchen table.The base of the saltshaker forms a regularoctagon. The eight sides ofthe octagon are markedwith red lines and yellowcongruency markings, andthe angles are markedwith blue. Definitions: A regular polygon is a polygon (a closed plane figure formed by three or more segments) that is both equiangular and equilateral. 17
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Garden Gate – Special Right Triangles (L. 53, L.56)This is a picture of a gate that myfather had built surrounding hisgarden to protect it from animalsthat would eat his plants.The right triangles in the gate are45-45-90 triangles, classified by themeasurements of their angles. Thelegs of these triangles arecongruent. The hypotenuse is thelength of the legs multiplied by √2.The right triangle to the right of thegate is a 30-60-90 right triangle,also classified by the measure of itsangles. The length of the smallestside (x) is doubled (2x) to get thelength of the hypotenuse. To find the 18length of the middle side, it ismultiplied by radical 3, (x √3).
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Floor Tiles – Tessellation (Investigation 9)This is a picture of a tiled floorhallway in my house.I have left this photographfairly untouched because thelines between the tiles arefairly distinct. I have, however,outlined the perimeter of thehallway lightly in black.The squares in picture form avery simple regulartessellation. Definitions: A tessellation is a repeating pattern of plane figures that completely covers a plane with no gaps or overlaps. 19 A regular tessellation is the simplest kind of tessellation, a repeating pattern of congruent regular polygons.
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Geometry in Nature – Triangular Dog’s Head (L. 51)This is the head of my greyhound,Opal.Her head is shaped like an acuteisosceles triangle, Triangle QRS. <Qand <R have congruent measures,while the vertex angle, <S does not.This is because the angle across fromthe line segment that is notcongruent to the other two linesegments will not be congruent to the 20other two angles of the triangle. InTriangle QRS, Line Segment QS andLine Segment RS, the legs, arecongruent, while Line Segment QR is Definition:not.T h e o r e m 5 1- 1:Is o s c e le s An acute triangle is a triangle with three acute angles (angles with measures lessT r ia n g le than 90°)T h e o r e m – If a An isosceles triangle is a triangle with at least two congruent sides. 20t r ia n g le is A vertex angle is the angle formed by the legs of the triangle.is o s c e le s , t h e n
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Geometry in Nature - StarfishThis is a starfish I found on atrip to the beach last summer. Ithen took it home and createdan ornament using a nail and apiece of string.A starfish has five legs, each legwith two sides. This gives it atotal of ten sides, making it adecagon¹. 21 ¹ See slide # 4 for a definition of “decagon.”
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Geometry in Nature – Spherical PlantsThis plant is called an Allium, which,when bloomed, is a sphere of purpleblossoms. My mother plants manyAlliums in her garden at home, andthis is where I found it.From the plant, you can see thecenter from which the blossomsstems has been labeled as thecenter of the sphere (Point A), andtwo of the buds have been labeled aspoints (Point C and Point B). Thesebuds represent points on thesphere’s exterior. D e f in it io n s : A s p h e r e is a s e t o f p o in t s in s p a c e t h a t a r e a f ix e d d is t a n c e f r o m a g iv e n p o in t c a lle d t h e c e nte r o f the s p he re . 22
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Bibliography• Saxon Geometry. Student ed. Austin: HMH Supplemental Publishers Inc., 2009. 800-871. Print.• "circle." Dictionary.com Unabridged. Random House, Inc. 11 Jun. 2012. <Dictionary.com http://dictionary.reference.com/browse/circle>.• Simmons, Bruce. "Frustum of a Cone or Pyramid." Mathwords. N.p., March 24, 2011. Web. 11 Jun 2012. <http://www.mathsisfun.com/quadrilaterals.html>.• "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram." MathIsFun.com. N.p., 2012. Web. 11 Jun 2012. <http://www.mathsisfun.com/quadrilaterals.html>.• . "Rectangular Prism - Geometry - Math Dictionary." icoachmathc.com. High Points Learning Inc, 199-2011. Web. 12 Jun 2012. <http://www.icoachmath.com/math_dictionary/Rectangular_prism.h tml>. 23
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