Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using te formula A ( ) b Find a missing side of a triangle using properties of proportion and similar figures
Autors: Jason Marc, B.A. Tim Wilson, B.A. Editor: Grapics: Linda Sanks Tim Wilson Jason Marc Eva McKendry National PASS Center BOCES Geneseo Migrant Center 7 Lackawanna Avenue Mount Morris, NY 40 (8) 68-7960 (8) 68-7969 (fax) www.migrant.net/pass Developed by te National PASS Center under te leadersip of te National PASS Coordinating Committee wit funding from te Region 0 Education Service Center, San Antonio, Texas, as part of te Matematics Acievement Success (MAS) Migrant Education Program Consortium Incentive project. In addition, program support from te Opportunities for Success for Out-of-Scool Yout (OSY) Migrant Education Program Consortium Incentive project under te leadersip of te Kansas Migrant Education Program.
Your friend, Jorge, comes up to you all excited about is new job at te airport. You ask im wat e does and e says, I cut te grass between all te runways. You reply, Tat sounds ard. Tat s a lot of grass to cut. Jorge says, I get paid ten cents for eac square foot I cut. You exclaim, Tat s great! How many square feet are tere between te runways? He replies, I ave no idea. Te runways form a triangle. Jorge ten sows you a diagram of te runways. You decide to elp im figure out ow muc money e will make cutting te grass. Jorge explains tat Runways and 3 are 00 feet long and form a rigt angle. Runway is te longest at 4 ft. You decide to redraw tis using a rigt triangle. 4 ft. 00 ft. Tink Back 00 ft. A rigt triangle is a triangle wit one rigt angle. Te only area formulas tat t you ave learned so far are for squares and parallelograms. You decide to figure out te area using tese familiar sapes. You notice tat te two sides of equal lengt form a rigt angle. Tis reminds you of a square, so you take a square and line it up wit te triangle. 4 ft. 00 ft. 00 ft. Mat On te Move Lesson 0
You see tat te two sides of te triangle line up perfectly wit te square. You also notice tat te longest side looks as if it cuts te square in alf. You can see tat te area of te triangle is one-alf te area of te square. Te area of te square is base times eigt. Area b (00 ft.)(00 ft.) 0,000 sq. ft. If te area of te triangle is one-alf of tat,, ten Area of triangle, 000 sq. ft. 00 ft. So, Jorge will make,000 $.0 $00 for cutting te grass. 00 ft. You discovered d tat te given triangle was made by cutting a square in alf. In fact, all triangles can be made by drawing te diagonal of a parallelogram. A diagonal of a polygon is any a line segment, oter tan a side, tat connects two vertices. In te following parallelogram, two diagonals can be drawn. Eac of tese diagonals cuts te parallelogram in alf, forming two triangles. Since te diagonal cuts te parallelogram in alf, te area of eac triangle formed is one-alf of te parallelogram. If te t area formula for a parallelogram is A b, ten te area of a triangle is Area of atriangle b Mat On te Move
Tese are te two triangles formed by te parallelogram. Tey ave te same area, because tey cut te same parallelogram in alf. Tese triangles sare te same base and eigt. obtuse angle b b Te eigt of te triangle is te perpendicular line segment drawn from te base to te vertex opposite te base. Notice tat te eigt of te above left triangle is drawn outside te triangle. In order to draw te eigt of an obtuse triangle, you must extend te base as sown above. Example Find te area of te following triangles. a) b) c) 8 in. 4 ft.. in. 6 ft. Solution To answer tese problems, we need to remember tat te area formula for a triangle is A b Tink Back Wen variables and numbers are written next to eac oter, it means multiplication. Mat On te Move Lesson 0 3
a) Tis is a rigt triangle wit a base of. units and a eigt of unit. So te area is A (. )( ). square units. Don t forget to include units in your answer. b) In tis triangle, te base is in. and te eigt is 8 in. Te area is square units units square feet feet A ( )( 8 ) 48 square inces. c) In te last triangle, te base is 6 ft. and te eigt is 4 ft. Te area is A ( 6 )( 4 ) square feet. Example Find te missing eigt of te triangle. Area 6 Solution 8 We are given te lengt of o f te base, as well as te area. We need to find te eigt. If we recall our formula for te area of a triangle, A b, and we substitute te dimensions we are given, we get, 6 8 6 4 6 4 4 4 4 Mat On te Move Multiply numbers first. Divide to get te variable all alone. 4 Ceck: 4 6 8 6 8 6 8 6 3 6 6 ( ) ( 4)
Find te area of te following triangles. ) ) 9 cm 0 cm 7 3) 4) 6 km 7 km 4 m Find te missing base or eigt of te following triangles using te formula A b. ) Area 0 m 6) Area units 8 m b 0 Mat On te Move Lesson 0
You talk wit Jorge a few weeks later. He says tat e was offered ed a different job at te airport. Now, tey want im to edge te grass around te perimeter of te runways. He will earn two dollars for eac foot of grass e edges. He wants you to elp im figure out ow muc money e will make compared to wat e earned cutting te grass. Finding te perimeter of a triangle is te same as finding te perimeter of a quadrilateral. All we ave to do is add up te lengts of all te sides. In te triangle made by te runways, te perimeter is te sum of all te sides. s. 4 ft. 00 ft. Te perimeter is 00 + 00 + 4 34 ft. 00 ft. Tis means tat Jorge will make 34 $ $684, for edging te grass., for edging te grass. Comparing $684 wit $00 (te money e earnrd rd mowing), 684 00 $84 Jorge can see tat e will make $84 more edging te grass tan cutting it. Example Find te perimeter of te following triangles. a) b) c) 0 mi. 8 ft. 3 ft. mm 6 ft. 4 mi. Solution To find te perimeter of eac of tese triangles, we ave to add up all te sides. Mat On te Move 6
a) We are given te lengt of all te sides in tis triangle. Te perimeter is 6 + 3 + 8 37 ft. b) We are only given two sides in te isosceles triangle. We know te tird side is 0 mi., because te as mark sows us tat te unknown side is congruent to te side tat is 0 mi. 0 + 0 + 4 34 mi. c) We are only given one side of te equilateral triangle. All te sides are equal in an equilateral triangle, so all te sides are mm. + + 33 mm Example If te total perimeter of te triangle is 8 cm, c, find te lengt of te missing side. 8 cm a 8 cm Solution Since te total perimeter is 8 cm, we know tat te sum of te lengts of te sides is 8 cm. Tis means we can set up te equation, 8 + 8 + a 8 6 + a 8 6 6 a Ceck: a 8 + 8 + a 8 ( ) ( ) 8 + 8 + 8 8 + 8 + 8 8 8 Mat On te Move Lesson 0 7
Find te area and perimeter of te following triangles. 4 m 3 7) 8) m m Given te following perimeters, find te lengts of te missing side(s) of te following triangles. 7 ft. 9) 3 ft. 0) Perimeter 7 m. z x x Perimeter 8 ft. 3 m On Wednesdays, you volunteer at a local yout center were you mentor younger kids. One yout tat you mentor, Ivette, needs some elp. Se explains. My Uncle Hernandez races sailboats. Te boat e sails is called a sloop. It is great for sailing upwind. He took me out on is sloop tis summer, and I loved it so muc tat I wanted to build a scale model of it. Te only problem is tat my broter took ok te model out of te package and lost a few of te parts. Can you elp me build new parts? Sloop sailboat Mat On te Move 8
Se continues, Te main sail of my uncle s sloop as tese measurements: Ivette sows you a diagram se drew. I only ave one part of te main sail left. Ivette asks you, How long sould I make te oter piece of my main sail? 30 ft. ft. ft. Uncle Hernandez s sloop Ivette s sloop model You realize tat since te model as been made to scale,, its sails and te actual sails of te sloop will form similar triangles. You draw a simplified version of Ivette s diagram. Te lengt of te bottom piece is not known, so you use a variable, x,, to represent it. 30 ft. ft. ft. Uncle Hernandez s sloop x Ivette s sloop model Mat On te Move Lesson 0 9
Tink Back If two figures are similar, te lengts of teir corresponding sides are proportional. Triangles wit tree congruent angles are similar triangles. (Tis is only true for triangles.) Since te figures are proportional, te ratio of te lengts of teir sides will be equal. Tis means, big bottom big side x 30 little bottom little side x 30 30 x 4 30 30 x 0.8 Tink Back Cross multiply. Wen given a proportion, cross multiply, ten divide. Divide eac side by 30. You tell Ivette, you sould make a piece tat sticks out eigt- tents of a foot. Tank you so muc! se exclaims. You are te best! Mat On te Move 0
Example In tis figure, AD 30. Wat is te lengt of DE? A B 3 C D E Solution How many triangles do you see in te diagram? You sould see two: ABC and ADE. Let us redraw tem below, wit te measurements given. We will use a variable, a,, to represent te side tat we are trying to find, DE. A A 30 D a E B 3 C Mat On te Move Lesson 0
Next, we will set up our proportion. Note tat tere are many ways to set up proportions wit similar figures. We will use te corresponding parts. Tat is to say, big side big base little side little base AD AB 30 DE BC a 3 90 a Now substitute te lengts for eac side. Note: Tere are a few ways to solve tis equation. Since cross multiplying always works, we will use tis metod. However, if you see a quicker way of solving tis (and tere is one), ten feel free to use tat. Just make sure you ceck your answer. 8 a Ceck: a 8 30 a 3 30 30 6 ( ) 3 6 6 ( 8) 3 8 3 Mat On te Move
Example Elena owns a landscaping business. One day, se gets a call from a customer wo wants a tree cut down. Se gets to te job site and needs to know ow tall te tree is for billing reasons. Se knows er own eigt is feet. Te sun is also casting te sadow of er and te tree. Find te eigt of te tree. 4 ft. 7 ft. Solution Wit a little imagination, we can redraw te diagram above using triangles. We label te information given and use a variable,, to represent wat we need to find. ft. 4 ft. 7 ft. Tese triangles are similar, so we can set up a proportion. Mat On te Move Lesson 0 3
eigt of tree eigt of person sadowof tree sadowof person 7 7 4 7 4 7 0 7 30 Ceck: 30 ( ) ( 30) 4 7 4 7 4 7 4 6 7 6 6 Mat On te Move 4
Given te similar triangles, use proportions to solve for te missing side. (Round te te nearest tent.) ) 6 x ) 7 p 9 4 3) m 8 m 7 m Mat On te Move Lesson 0
Review. Higligt te definition of diagonal.. Higligt te Tink Back boxes. 3. Write one question tat you would like to ask your mentor, or one new ting you learned in tis lesson. Practice Problems Mat On te Move Lesson 0 Directions: Write your answers in your mat journal. Label tis exercise Mat On te Move Lesson 0, Set A and Set B. Set A Use te formula A b to find te area of eac triangle to te nearest tent. ) ) 3) 3 A triangle wit a base of 7 feet, and a eigt of feet 4 7 Mat On te Move 6
Use te formula, A b to find te missing base or eigt of eac triangle. 4) ) 4 in. Area 0 in. Area 48 ft. b b ft. Find te perimeter of eac triangle. 6) 7) 8) m 9 m ft. 9 mi. m 3 ft. Given te perimeter, find te lengt of eac missing side of eac triangle. 9) 0) 3 cm 9 mi. 8 mi. x x z Perimeter 7 mi. Perimeter 4 cm ) Perimeter 78 Mat On te Move Lesson 0 7
Given similar triangles, find s. ) s 0 3 3) 0 3 s Set B ) Draw two polygons wit corresponding congruent angles, tat are not similar. ( (Hint: tey will not be triangles.) ) For te following problem: Find z. 8 4 7 z Melissa sets up te following proportion: z 7 4 8 Hector sets up tis instead: 7 z 8 4 Wo is correct, Melissa, or Hector? Solve for z in bot equations. Wat do you notice? Wy do you tink tis is? Mat On te Move 8
3) Te perimeter of te following triangle is 6. Find te lengt of eac side. x x x + ) A 4 cm ) A 7 units 3) A 8 m 4) A km ) 0 8 b 6) 0 b m units 7) A 30 units 8) P 30 units P 30 m A m 9) z 8 ft. 0) x + 3 7 ) x units ) p. units x 7 m 3) 3. 6m Mat On te Move Lesson 0 9
NOTES End of Lesson 0 Mat On te Move 0