Corresponding angles must be. Corresponding sides must be. Basically: Same shape, different size.

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1 Similarit What makes two polgons similar? Geometr 11.2 orresponding angles must be. orresponding sides must be. asicall: Same shape, different size. re both definitions necessar? Tr to draw two polgons with congruent angles that are NOT similar. Tr to draw two polgons with proportional sides that are NOT similar. Similarit can be used to find missing sides and angles of polgons: Solve for a, b,,, and z in the similar pentagons below. 50 o 95 o a 25cm 45 o b 95 o Triangle Similarit ecause triangles are rigid, there are similarit shortcuts for triangles (Just as there are congruence shortcuts... name them all.) In triangles, if all three angles are equal, the sides will alwas be proportional. onsider a right triangle. Is it possible to change the length of one side and maintain all three angle measures? How man angles do ou need to determine similarit? 6cm Since is a similarit shortcut, so are, S, and S. SS is a similarit shortcut. SSS is also a shortcut. onsider again two right triangles for demonstration: right triangle and a right triangle. If all three sides are proportional, two triangles are similar. SS is not a shortcut for congruence, and cannot be used to prove similarit either because multiple triangles can be drawn with SS similarit. () is the onl similarit shortcut that is not a congruence shortcut. o z o 18cm 15cm o

2 Similarit Shortcuts Geometr 11.2 Practice: etermine which pairs of triangles below can be proven similar and b which shortcut: 10cm 60 o 75 o 15cm 24cm 18cm 18cm 34 o 72 o 18cm 18cm 60 o 45 o 34o 45 o 36cm 52cm J 20cm 26cm 34 o 74 o H 20cm 27cm 24cm F 39cm 34 o 10cm 16cm G 14cm K 15cm 12cm 45 o 5cm b, G b SSS, H b SS, not to F (SS), J K have no partner Practice: Name all the similar triangles in each of the diagrams below: Y 92 o J H 36 o G 5.5cm F 4cm W 4cm V R 9cm 6cm 4cm 4cm 5.5cm T Q 7cm 7cm S Z

3 Indirect Measurement Geometr 11.3 Shadow proportions: 1. The Sears Tower in hicago is our nations tallest skscraper. If the Sears tower casts a shadow that is 1,160 feet long while a parking meter that is 5 feet tall casts a 48-inch shadow, how tall is the Sears Tower? 2. Nick s shadow is 5 inches longer than Smirthi s shadow, but he is 7 inches taller than she is. If Smirthi is 59 inches tall, how long is her shadow? (to the tenth) Sight Lines and istant Objects 1. You are in a Washington state park at a sign that points out major mountain peaks in the distance. It includes arrows pointing to Mount Rainier 72 miles awa, and Mount St. Helens 55 miles awa. You just happen to have three metersticks available. plain how ou could use them to determine the distance between the two peaks. 2. You and a friend want to know the eact distance across a large lake. You use stakes, string (dark lines), and sight lines (dashed) to create the following setup: What could ou do to make and parallel? What measurements would ou need to determine the distance across the lake? Practice: 1. You want to know the height of a flagpole (but it is cloud). You hold a 6cm ruler verticall 45cm in front of our ee so that the top of the flagpole aligns with the top of the ruler, and the base of the flagpole aligns with the base of the ruler. What additional measurement(s) can be made to determine the height of the pole? 2. Uisng stakes and a tape measurer, eplain how ou could determine the eact distance across a river to a tall tree on the opposite bank. 10m 28m 35m

4 orresponding Parts Geometr 11.4 orresponding Parts of Similar Triangles: If a pair of triangles is similar (corresponding sides proportional) then the corresponding altitudes, medians, and angle bisectors will also be proportional. Solve: Solve for the length in each pair of similar triangles below 1. ltitudes. 2. Medians F H How can ou show ~ FH? 30 G L 5 M Q 6 R P 11 9 T 13.2 N How can ou show LMP ~ QRT? S Solve: Solve for the length in each pair of similar triangles below. 1. ngle bisectors. 2. Solve for,, and z. K W Y L M N Z 3 5 How can ou show WZY ~? P 2.8 Q Name all pairs of similar triangles. z R

5 ngle isector Theorem ngle isectors Geometr 11.4 ngle bisectors do not bisect the opposite side of a triangle, but there is a special propert of angle bisectors ou should understand: ngle bisectors in a tiangle divide the opposite side into segments whose ratio is proportional to the adjacent sides: = Proof: dd point so that. This creates similar triangles and such that Some simple work with the angles shows that = = =, which means that triangle is isosceles and =. Substitution ields: = = Practice: Find the missing lengths in the figures below: Round to the hundredth F 9 G 21.6 H 7 L K p N 25 r M W Z Y 5 a 4

6 ngle isector Theorem Geometr In each diagram, the cevian (line from a verte to the opposite side) is an angle bisector. Find the missing length in each diagram. Round decimal answers to the hundredth or leave them in radical form cm 3cm 6cm 8cm 13cm 3. (isosceles triangle) 4. ( is the length of the angle bisector) 12cm 18cm 25cm 7cm 5. (The heagon is regular. Find.) 6. ( is the length of the angle bisector, round to the thousandth) 3cm 3cm 5cm 3cm 7cm

7 ngle isector Theorem Geometr Find the missing lengths in each diagram. Round decimal answers to the hundredth or leave them in radical form. iagrams not to scale. 7. In triangle, =14, =15, and =16. Point is on such that bisects angle. Find the length of. 8. Square FGH has side length 6cm. J is the midpoint of side GH. If FH and J intersect at, what is the length of J? F H J G 9. In right triangle TUV, TV=3cm and UV=4cm. ngle bisectors UW and TY intersect at. Find the length of segment U. T a. Find WV and WT. b. Find UW (in radical form). c. Find U. U a 5 Y W V hallenge. The hpotenuse of a right triangle is the angle bisector of angle M in triangle KLM. Find the length of MN. press our answer as an improper fraction in simplest form. (If ou do this correctl, the denominator will be 119). M L N 5 K

8 Right Triangles Geometr 11.4 ltitude to the hpotenuse of a right triangle: Name three similar triangles in the figure above. amples: Find the missing lengths in the figures below. Round decimal answers to the hundredth. 4cm z 3cm 4cm z 9cm Practice: Find the missing lengths in the figures below. Round to the hundredth R 70cm F 67.2cm z P 3cm N 7cm z T hallenge: 12cm 6cm

9 Other Useful Similar Triangles Two more common tpes of similar triangles: Name the pairs of similar triangles in the diagram below. Order is VRY important here. Look at which angles are congruent. Geometr 11.4 T Z V U W Use what ou know to find the missing length below: Leave answers as fractions in simplest form. 6cm 8cm 3cm 7cm 8cm 4cm 5cm More ifficult Practice: Solve. 1. In rectangle, is the midpoint of. a. Find the length of. b. Find and The diameter of circle is 22. =8 and =12. Find.

10 Parallel Lines: Similarit Solve: Find the missing lengths below. Geometr cm 10cm 18cm 15cm a 6cm 24cm b 28cm Solve: Write as man proportions as possible which relate the lengths of sides listed below: b c d w s t a z The converse also works. If corresponding lengths between lines are proportional, the lines are parallel. Solve: Which of the lines below are parallel? a b w cm 18cm 10cm 15cm c z 9cm 10cm 8cm 12cm 12cm 12cm 18cm 24cm

11 24cm Similarit in Right Triangles Geometr 11.3 Find the missing lengths in each diagram. Round decimal answers to the hundredth or leave them in radical form. iagrams not to scale cm 16cm 10cm 9cm Leave #1 in radical form (hard: Use Quadratic Formula) 4cm 2cm 20cm 25cm Find the diameter of the circle: 6. Find the area of square F. 31.2cm 31.2cm F diameter = 6. area =

12 Similarit Solve. iagrams not to scale. Geometr is a diameter. =15 Find (>) M 5cm 4cm cm is an isosceles trapezoid. ==56cm Find the area of triangle. Leave our answer in simplifed radical form. 54cm 8.

13 Power of a Point: ase 1: Two chords. Find the missing length : Geometr 5cm 2cm 3cm R M ase 2: Two secants. Find the missing length : 8cm 7cm 9cm ase 3: Tangent/Secant. Find the missing length : 7cm 4cm Practice: Find the missing length (not to scale): 7cm 10cm 4.8cm 6cm 3cm 16cm 9cm 11cm

14 Similarit Solve. iagrams not to scale. Geometr 1. n isosceles triangle with congruent side length of 5 and base 6 is inscribed within a circle. What is the diameter of the circle? 2. You are estimating the height of a semicircular arch. actl 1 foot from the base of the arch, the arch is 5 feet tall. What is the height of the arch at its center? 5 ft 3. What is the area of the circle below (in terms of Pi)? 2cm 8cm

15 Similarit Solve. iagrams not to scale. Geometr 4. What is the combined area of the five squares below if =3cm and =6cm? 5. Find the length of in the diagram below if = = 3 and = What is the area of the right trapezoid below if = 8, and = 12?

16 rea and Volume: Similarit hanging dimensions of similar figures. amples: Geometr square has 3-inch sides. larger square has 6-inch sides. How man times larger is the area of the large square than the smaller one? 2. cube has 3-inch sides. larger cube has 6-inch sides. How man times larger is the volume of the large cube than the smaller one? 3. circle has a 3-inch radius. larger circle s radius is five times as long. How man times greater is its area of the large circle than the small one? 4. The volume of a clinder is 15cm 3. What would be the volume of a clinder with 3 times the radius and height? Similar figures have the same shape and different size. For eample, right cones can be similar if their heights and radii are proportional. The ratio of the lengths of sides in simlar figures is called the scale factor, for eample the scale factor for the pentagons below is If similar two-dimensional figures are increased (or decreased) b a scale factor of n, then the resulting area will change b a factor of n 2. If similar three-dimensional figures are increased (or decreased) b a scale factor of n, then the resulting volume will change b a factor of n 3. Solve: 1. regular pentagon with 5-inch sides has an area of approimatel 27.5 square inches. What would the area of a regular pentagon with 10- inch sides be? What about a regular pentagon with 7-inch sides? 2. Similar cones are 3-inches tall and 5-inches tall. What is the ratio of the volume of the larger cone to the volume of the smaller one?

17 rea and Volume: Similarit Solve: Geometr What is the ratio of the area of the square on the right to the area of the square on the left? Write our answer as both an improper fraction and a decimal or 2. Squares have perimeters of 12 and 18 inches. What is the ratio of the area of the larger square to the area of the smaller one? Similar triangles have perimeters of 24 and 36 inches. The area of the smaller triangle is 24in 2. What is the area of the larger triangle? hint: It is NOT 36in 2. Use the scale factor. Onl use the diagram below if ou have to in Similar polgons have areas of 18in 2 and 8in 2. The larger polgon has a perimeter of 30 inches. What is the perimeter of the smaller polgon? hint: use the scale factor of their areas to find the scale factor of their perimeters cube has edge length 4. The edge length is increased to 10. 5a. what factor has the surface area increased? 5b. what factor has the volume increased? 5a. 5b. 6. ube has edge length 3. What is the edge length of cube if it has twice the volume of cube (to the hundredth) Tetrahedron has edge length 5. What is the edge length of tetrahedron if it has twice the volume of tetrahedron (to the hundredth). 7.

18 rea and Volume: Similarit Geometr 11.5 With similar figures and, if ou know the ratio of the side length a/b then the ratio of their areas is (a/b) 2 and the ratio of their volumes (if applicable) is (a/b) 3. Ratio: Sides/ rea Volume Perimeter a b a b 2 a b 3 ample: The edges of a regular tetrahedron are 6cm long. The surface area is about 62cm 2 and the volume is about 25cm 3. pproimate the surface area and volume of a tetrahedron whose edges are 7cm long. rea = (62) cm 2 Volume = (25) cm 2 Practice: 1. The area of a circle is 5cm 2. If the radius is tripled, what will be the area of the new circle? The perimeter of a regular heptagon is 19 inches and its area is approimatel 27 square inches. pproimate the area of a regular heptagon whose perimeter is 24 inches. (to the tenth) The volume of a sphere whose radius is 1.85cm is about 26.52cm 3. pproimate the volume of a sphere whose radius is twice the original (3.7cm). heck our answer using the volume formula for a sphere regular octahedron has 12cm edges. Its volume is approimatel 815cm 3, and its surface area is about 499cm 2. pproimate the surface area and volume of a reguar octahedron with 8cm edges to the nearest whole number. 4. Vol. S..

19 rea and Volume: Similarit Geometr 11.5 Mied Review: Solve. Round decimal answers to the hundredth. 5. In right triangle, =3 and =5. ngle is bisected b, with on. Find the length of angle bisector In right trapezoid, diagonals and intersect at. =12, =20, and =37. Find the length of to the nearest hundredth When cut in half, a sheet of paper ields two smaller rectangular sheets which are similar to the original sheet. If the original sheet is 4 inches tall, how wide is it? (leave in radical form) 4 8. In the triangle below, is four times the length of. If the area of triangle is 20cm 2, what is the area of triangle? The volume of a regular icosahedron with 5-inch edges is about 270in 3. What is the edge length of a regular icosahedron whose volume is 10in 3? 9.

20 Little lues rea and Volume: Similarit Using scale factor and similarit: Geometr 11.5 amples: The area of a triangle with side lengths 5, 7, and 10 is What would be the area of a triangle with side lengths of 50, 70, and 100? Mr. atterson has his own action figure. It looks eactl like him, but it is just 11.5 inches tall. Mr. atterson is 5-9 and weighs 162 pounds. If the action figure is the same densit as Mr. atterson, how much does it weigh? Practice: 1. You are painting a mural onto our school based on a drawing ou created on a sheet of paper that is inches. The mural will be 34 b 44 feet. The mascot portion of the mural is 10in 2 on the sheet of paper. What will the area of the mascot portion be on the actual building in square feet? (hint: the answer is between 100 and 200 ft 2 ) L M S n artist carves three perfectl similar cones out of marble. The smaller of the three cones is 3 feet tall and weighs eactl one ton (2000 lbs)! The other pramids are 40 inches and 44 inches tall. How much more does the largest cone weigh than the smallest cone (to the pound)? 2.

21 rea and Volume: Similarit Geometr 11.5 Practice: 3. restaurant in Ital is tring to break the world record for the largest pizza ever baked. To break the record, the need to bake a pizza with a 126 foot diameter. Their normal pizzas have onl a 14 inch diameter and it takes a pound of cheese to cover the pizza. How man pounds of cheese will the need to cover the giant pizza? Your mom has a garden statue that is an eact replica of the statues at aster Island. He garden statue is just 2 feet tall. The real statues average 14 feet tall and weigh 27,440 pounds. If her stone replica is made of the same stone as the real moai statues, how much does it weigh (in pounds)? rass east trophies are handed out for Mr. atterson s math class. ach is solid brass and the third place troph is 15 inches tall. Mr.. wants the first place troph to be eactl three times as large (b mass) as the 3rd place troph. To the tenth of an inch, how tall should it be? The surface area of a solid cla hemisphere is 10cm 2. larger solid cla hemisphere has a surface area of 40cm 2. If the larger hemisphere weighs 2 pounds, how much does the smaller one weigh? 5.

22 Similarit Practice Geometr 11.7 Solve: ach of the problems below can be solved using proportional reasoning. Working with a partner, solve as man as ou can. 1. The top right corner of a sheet of paper is folded down and left to align with the left edge,and the bottom right corner is folded up and left so that the fold lines look like the diagram below. What is the ratio of the area of the large isosceles triangle to the smaller triangle? 2. and are chords of a circle which intersect at. If = 3, = 4, and = 2 what is the length of? 3. Find the length of chords,, and. 60cm 40cm

23 Similarit Practice Solve: Geometr ircle has a radius of 28 inches, and circle has a 44-inch radius. The distance between their centers is 90 inches. What is the length of segment, which is eternall tangent to both circles? 5. In each right triangle below, solve for lengths,, and z. Leave answers in radical form or round to the hundredth. F z z 7cm H 4cm G 6. sphere is inscribed in a cone whose radius is 6cm and whose height is 8cm. Find the volume of the sphere. r

24 rea and Volume: Similarit Geometr Practice: 1. The edge length of a cube is doubled. The original cube had a surface area of 96cm 2. What is the surface area of the larger cube? 2. The edge length of a triangular prism is doubled. The original prism had a volume of 10cm 3. What is the volume of the larger prism? rectangular prism has each of its edge lengths increased b 40%. what percent is its volume increased? 4. Similar cones have volumes of 10cm 3 and 80cm 3. If the height of the smaller cone is 10cm, what is the height of the larger cone? n icosahedron has each of its edge lengths increased b 30%. what percent is its volume increased? Similar pramids have surface areas of 50cm 2 and 80cm 2. What is the ratio of the volume of the smaller pramid to the volume of the larger one? press our answer as a decimal to the nearest hundredth cube has an interior diagonal length of 5 3 cm. larger cube has twice the surface area. What is the length of its interior diagonal? Michael has a scale model of a Ford Mustang. It takes a proportional amount of paint to coat the model as the real mustang. If the model is 1/12 the length of a real Mustang and requires 2 ounces to paint, how man gallons will it take to paint a real Mustang (128oz = 1 gal.) 8.

25 Similarit Practice Test Solve: Geometr t 3pm, Sam s shadow is 6 inches taller than Mar s, even though he is onl 4 inches taller than she is. If Sam is 9 inches taller than llison, how much longer is Sam s shadow than llison s? To determine the width of a lake, the measurements below are taken. What is the width of the lake? (note: ) = 32 feet. = 22.4 feet = 42 feet 2. Solve: Find the missing lengths below ft 15 ft 8 ft 12 ft 10 ft 40cm 24cm 9cm 30cm 3. = 4. = Solve: Find the missing lengths below. Round to the hundredth cm 4cm 9cm 8m 17m 5. = 6. =

26 Similarit Practice Test Solve: o not round our answers. Geometr Similar pramids have altitudes of 3cm and 4cm. The volume of the smaller pramid is 30cm 3. What is the volume of the larger pramid? t an art museum, there is a 14-foot tall solid marble statue of a horse. t the gift shop ou bu a solid marble replica of the statue that weighs about 2 pounds and is onl 8 inches tall. stimate the weight of the full sized original statue Find the length of chord in circle R below. (otted lines are hints, fill-in everthing ou know and ou ll get there.) 12cm R 5cm What is the area of isosceles triangle below to the hundredth? 9cm 4cm 10..