Ratios and Proportions
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1 Warm Ups Ratios and Proportions Ratio: expression that compares_ two quantities by division Examples: Proportion: a statement of _equality between two _ratios Examples: Solve examples of proportions with x involved: x b a a X b a 20 a c 3c d 4d + 8 f c 12 d 8 f -5, 5 c 2 Word Problems- write the proportion and solve. 7. If eggs cost $2.04 per dozen, how much would 5 eggs cost? 2.04 x x 10.2 x The ratio of two numbers is 5:2_. The smaller number is 50. Find the larger number. 5 x x 250 x 125 1
2 Similar polygons: Two polygons with Corresponding angles that are _congruent Corresponding sides that are _proportional If these two polygons are similar, we write a similarity statement : _ABCD~EFGH 9. Write a similarity statement: _ ABCD~EFGH x y z 10. Given that ABC~DEF. Find the following: y 75 12y 6.25 y z 45 12z 3.75 z 12 x x 7.2 x 30 o 140 o Solve each proportion: a. 2 x x b. c x 96 x x x x x X 16 x 1.4 x 8 8x 39 X e. 5 y w 1 4 f. 56 g x 2
3 15 8y 2w x x y w 3.2 x 3 x 5 i. 2 x a j. k. 3 x 1 2 x + 4 l. 3 x 6 4 x a x a + 4 3a 6 15x 5 8x x 24 8x x 4 a 6 7x x x 10 a 7x 21 4x 20 x 3 x 5 o 13. The two triangles below are similar, ABC ~ DEF. m A 42, m B 58 o x y m D 42 o B 7 cm 8 cm C 6 cm A E 3 cm x F D y y x 18 7y 2.57 y y x 24 7x 3.43 y m F 80 o 14. ABCDE ~ FGHIJ. X E w _36 x 45 y 27 6 G F J H 9 12 I Z B A 54 C W Y D z 18 Proportionality Statements Warm Ups 3
4 Learning Objectives I can find missing measures in similar figures. Given, set up a proportionality statement. 1.. Find the length of BE and BA. 2.. Find the length of BC and AC. BC 6.8 cm AC cm 3. Write a proportionality statement and solve for y. 4
5 The _ratio of any two corresponding lengths in two similar geometric figures is called scale factor. Different ways that scale factor can be written: 4/3, ¾, 4:3, 3:4,.75, The blueprint for a house uses the scale factor of 4 in:7 ft. How many feet are represented by 1 in? 1.75 ft How many times larger is the house than the blueprint drawing? 4 in : 84 inches. 1 inch: 21 inches. The house is 21 times larger. 4. Given, what is the scale factor? Find the value of x. Scale factor: _3:7 5. Given, what is the scale 6. Find the variables given the two triangles factor? Find the value of x. are similar. 7. Success Criteria: Find the values of x and y. 8. ABDC ~ FGDE. Solve for x and y. 5 AB BD DC AC FG GD DE FE 5 BD y 16 2 GD 6 x 30 2 y 32 5x 15 y 6.4 x
6 9.. (a) Set up a proportionality statement. AB BC AC DB BA DA (b) What is the scale factor? 10:9 A 12 9 B C 10 (c) Find x and y x 9 y 81 10x y 8.1 x 10.8 y y D x 10. Triangle ABC is similar to triangle DBA shown below. Find the measure of DA and DB. Set up a proportion and solve for x. AB BC AC DB BA DA DB 3 DA 9 4 DB 18 4DA 2.25 DB 4.5 DA YX XZ YZ KJ JL KL 5 XZ 15 4 JL x 60 5 x 15x SQ QV SV SR RT ST SQ 5 x SR x 20x
7 KJ JL KL MP PL ML x 4 KL 12 6 ML 48 6 x 8x KJ JL KL MP PL ML x 4 KL 12 6 ML 48 6 x 8x Write a similarity statement for the two similar triangles and set up a proportion and solve for x. 15. YWU 16. AFC ~ _ DFB c c C C 24 WZ ZU WU YW UW YU 3x 6 x x x 12x AF FC AC DF FB DB AF DF 2x 1 2x x-4024x+12 16x 52 x Special Similarities Day 1 Warm Ups 7
8 Learning Objectives I can use knowledge of proportional medians, altitudes, and angle bisectors to find missing lengths. I can use properties of parallel lines and proportional segments to find missing side lengths. Explore: Use rulers to measure the distance of each side of the two triangles below. (a) Find the scale factor between the two triangles. (b) Find the midpoint of BC and EF. Draw the medians AM and DT. (c) Measure the lengths of these medians. Find the scale factor between them. (d) Draw the altitude from B to AC and the altitude from E to DF. Find the measure of each altitude and find the scale factor. Property 1: If two triangles are similar, then the corresponding _altitudes, corresponding medians_, and corresponding angle bisectors_ are _proportional to their corresponding sides. 1.. M and N are midpoints. Find h and j. 8
9 2.. Find a. Property #2: If a line parallel to one side of a triangle passes through the other two sides, then it divides them proportionally. Conversely, if a line cuts two sides of a triangle proportionally, then it is parallel to the third side. 3. Find the value of x. 4. Find the value of x and y. 5. Find the length of OP and NO. 9
10 OP NO 6. Find the value of x. (a) x 9 (b) x For each triangle, find the value of x. (a) x _24 (b) x For each triangle, find the values of the missing lengths. (a) (b) 10
11 2 3 4 w 12 2w 6 w z 9 2z 4..5 z m 36 8m 4.5 m n m n p 72 8 p 9 p q 81 8q q (c) (d) k k 21.6 k h h h 12 k k 15 k 11
12 Special Similarities Day 2 Warm Ups Learning Objectives I can use the triangle angle bisector property to find missing lengths. I can use knowledge of proportional medians, altitudes, and angle bisectors to find missing lengths. 1. (a) Construct altitude BD in triangle ABC. (b) Measure each of the sides below to the nearest tenth of a cm. CB BA AC CD AD (c) Are the three triangles similar? Why or why not? Will this always be the case when you construct an altitude from a right angle? Explain. 12
13 Measure segments a, b, p, and q to the nearest tenth of a cm. Can you find the equivalent ratios? Write a conjecture based on your findings. a b p q Property 2: The angle bisector in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the two sides forming the angle. 2. Solve for the measure of x. 3. Solve for the measure of x. 4. Solve for x and y. 13
14 5. Success Criteria. Draw the small, medium, and large triangles. Solve for the value of x x x x 6. Find the value of y. 7. Find the value of b. 30 y y 12 y 13 5 b b 7.8 b 8. Find the length of AB given BC 7.8 cm, AC 9.1 cm, AD 2.3 cm, BD 3.9 cm, CD 6.8 cm. AB _4.5 14
15 Solve for the missing variable. 9. x 10. AD 3 AD AD 2.4 AD 5.5 x x 11 x 11. CA AB CA _4.24_ 12. b b b 15b b 42b 756 b 18 15
16 Real World Problems Warm Ups Learning Objectives I can use similar figures to solve word problems. I can convert a measurement between different units. I can find the scale factor of similar figures and convert between units when lengths are given using different units. 1. A blueprint of a new addition for a school is modeled with a scale of 3 in: 4 ft. (a) If the bathroom is 8 inches by 12 inches on the blueprint, what are the actual dimensions? ft by 16 ft (b) How many times bigger is the actual bathroom? 16 times bigger (c) The scale factor would be 1 inch:16 inches or 1.33 ft 2. Mr. T is 6 ft tall and casts a shadow that is 7 ft long. He is standing next to a tree that is casting a 28 ft shadow. How tall is the tree? 3. Kareem needed a tree in his yard cut down. The tree company asked if he knew the height of the tree. Kareem wanted to find out. He started at the base of the tree and walked along the shadow of the tree until he noticed his shadow matched up the end of the trees shadow. He knew that he was six feet tall and the distance from his feet to the end of the shadow is 8 feet. Then, he measured the shadow of the tree to be 35 ft. Use this information to find the height of the tree. 6 x x x
17 4. Ms. Dahlke and Mr. Scardigli are building houses that are proportional in size. Mr. Scardigli's house is smaller than Ms. Dahlke's. With the information given find out how much longer Ms. Dahlke's roof is than Mr. Scardigli s roof. 5. A person 6 ft tall casts a 7 ft shadow. At the same time a nearby flagpole casts a 37 ft shadow. How tall is the flagpole? 6. Success Criteria: Mr. Hieb is 6 ft tall casts an 84 in shadow. How tall is Mrs. Holt if at the same time her shadow is one foot shorter than his? x x 432 x 5.14 ft 7. Two triangular roofs are similar. The corresponding sides of these roofs are 16 feet and 24 feet. If the altitude of the smaller roof is 6 feet, find the corresponding altitude of the larger roof x 16x 144 x 9 ft 17
18 8. Justus, who is 1.9 meters tall, wants to find the height of a tree in his backyard. From the tree s base, he walks 12.4 meters along the tree s shadow until his head is in a position where the tip of his shadow exactly overlaps the end of the treetop s shadow. He is now 6 meters from the end of the shadows. How tall is the tree? 1.9 x x x m 9. Mr. Hieb is 5 ft tall and casts a shadow that is 9 ft long. He is standing next to a 40 ft tree. How long is the shadow of the tree? x 5x 360 x 72 ft 10. A particular satellite is 15 meters wide. A model of the satellite was built with a scale of 3 cm : 5 m. (a) How wide is the model? 9 cm (b) How many times larger is the actual satellite? times bigger (c) Find the scale factor. 1 cm : _ cm 11. A model airplane is built with a scale of 1 in: 9 ft. The real plane has a wingspan of 35 feet. (a) Find the wingspan of the model. 1in xin 9 ft 35 ft 9x 35 x 3.88 in (b) How many times bigger is the actual airplane compare with the model? 1 in: 9 ft 1 in: 108 in 108 times bigger 18
19 Real World Problems Day 2 Warm Ups 1. Two ladders are leaned against a wall such that they make the same angle with the ground. The 10' ladder reaches 8' up the wall. How much further up the wall does the 18' ladder reach? 2. Two Triangles are similar. The sides of the first triangle are 7, 9, and 11. The smallest side of the second triangle is 21. Find the perimeter of the second triangle. 3. King Kong on top of the Empire State Building casts a shadow 120 meters long. The Empire State Building alone is 485 meters high and casts a shadow 97 meters long. How tall is King Kong? 4. Success Criteria: You are flying a kite and want to figure out how high it is. You tie the string to the ground and measure out 1 m and then measure that the string is 4 m high. The distance from where the string is staked to directly underneath the kite is 50 m. 19
20 5. Two ladders are leaned up against a wall at the same angle. The first ladder is 6 ft long and 3 ft away from the wall. The second ladder is 12 ft away from the wall. How long is the second ladder? x 72 3x 24 x 6. The sides of a triangle are 5, 6 and 10. Find the length of the longest side of a similar triangle whose shortest side is y 150 5x 30 x 7. A model train travels 12 kilometers/hour. How far does it travel in 10 minutes? (Hint: how many minutes are in an hour)? 12km xkm 60 min 10 min x 20 x 20
21 8. On the floor plan below, 2 cm represents 10 feet. ( (a) What are the actual dimensions of the master suite not including the walk in closet (WIC) and master bath? 3.5 cm by 3 cm Actual dimensions 16.5 ft by 15 ft (b) What are the actual dimensions of the combined kitchen and breakfast area? 2.8 cm by 4 cm Actual dimensions 14 ft by 20 ft (c) What are the dimensions of the dining area? 2.8 cm by 2.6 cm Actual dimensions 14 ft by 13 ft (d) How many times larger is the actual house compared to this floor plan? 2 cm : 10 ft 1 cm: 5 ft 1 cm: cm times larger Similarity Proofs 21
22 Warm Ups Learning Objectives I can prove if triangles are similar using similarity congruence theorems. Name the conjectures that prove triangles congruent. SSS, SAS, ASA, AAS, HL Name the conjectures that do not work when proving triangles congruent. AAA, SSA What are the differences between congruent triangles and similar triangles? Congruent Similar Same size, same shape Different size, same shape In order to prove two figures are similar, you must show: Corresponding sides are proportional Corresponding angles are congruent AA Similarity Theorem If _two angles_ of one triangle are _congruent_ to two angles of another triangle, then the triangles are similar This is what s called a _similarity statement_ 22
23 <L <M and <J 48 <Q <RSX <WST (vertical) and <X <T (alt int) JLK ~ QMP by AA SSS Similarity Theorem SXR ~ STW by AA If the _corresponding_ side lengths of two triangles are _proportional_, then the triangles are similar. SAS Similarity Theorem If the length of _two sides_ of one triangle are _proportional_ to two corresponding sides of another triangle and the _included angles_ are _congruent_, then the triangles are similar AB AC 3 and <A <A RT TS QP 2.5 AF AE 2 RQ QP RS ABC ~ AFE by SAS RQP ~ RTS by SSS Proofs 5. Given: DE AC 23
24 Prove: ABC ~ DBE Statements 1. DE AC Reasons 1. Given 2. B B 2. Reflexive property Corresponding angles are congruent 4. ABC ~ DBE 4. AA 6. Success Criteria Given: the segment lengths below Prove: IJK ~ Statements Reasons 1. (everything labeled in the figure) 1. Given IJ IK JK 2. TX TZ XZ 2. Corresponding sides are proportional (using the division property) 3. IJK ~ TXZ 3. SSS 7. Determine which triangle is similar to ABC by SSS. Write a similarity statement and find the scale factor ABC ~ DEF by SSS; scale factor 4/3 (or 3/4) 8. Determine if the two triangles are similar by AA. (a) ABE and ACD (b) DEC and GHK (c) CDE and BDA 24
25 <A <A and <ABE <ACE <D <G and <C <K <D <D and <BAD <CED yes yes yes 9. Determine if the two triangles are similar by SAS. (a) LNM and JNK (b) CDB and CEA 6 9 and <LNM <JNK (vertical) and <C <C yes 10. Determine whether the triangles are similar. If they are, state why we know that they are. (a) HGJ and HFK (b) yes yes, SSS yes, SSS (c) (d) SRT and PNQ yes, SAS yes, SAS (e) SVR and UVT yes, AA 11. Complete the proof. Given: 25
26 Prove: Statements Reasons Given 2. CBD CAE 2. Corresponding angles 3. C C 3. Reflexive property AA 12. Write a proof for the following. Given: NO 8 cm, OQ 10 cm MO 9 cm, OP cm Prove: Statements Reasons 1. NO 8 cm, OQ 10 cm 1. Given MO 9 cm, OP cm NO MO 2. QO PO 2. Corresponding sides are proportional POQ MON 3. Vertical angles are congruent SAS 13. Given: Prove: Statements Reasons Given 2. MNQ PNO 2. Vertical angles are congruent 3. M P 3. Alternate interior angles are congruent AA 26
27 Review Warm Ups Solve each proportion y y _ y y _ 2 3. The two triangles below are similar. Find 4. ABC ~ DBA. Find x and y. the scale factor of the values of x and y. DEF to ABC. Find Proportionality statement: o m C 85, m A 40 o C 12 Scale factor: _ 7 : 3 x _8.1 A 9 B 10 x _3.43_ y _2.57_ 7 cm C m F m E y _10.8_ y D x B 6 cm E 3 cm F y 8 cm A x D 27
28 5. Similar or not? YES or NO 6. GRE ~ PCU BC 2 ft, CE 8 ft, 6 2x x 2 DC 12 ft, and AC 3 ft. RE 12 x 5 If similar, ABC ~ _DEC_ By which conjecture _SAS_. CU _ 18 U 4x 2 C 9 G P 5 6 E 2x + 2 R 7. Find the value of x. x Given AB DE, write a proof (paragraph, two-column, or flow chart) to show that these two triangles are similar. Statements Reasons 1. AB DE 1. Given 2. A E 2. Alternate interior angles are congruent 3. ACB ECD 3. Vertical angles are congruent 4. ABC ~ EDC 4. AA 28
29 9. CBA has a segment CD that is an 10. AD is an angle bisector of BAC. angle bisector. Draw CBA. x x CB 12 cm, CA 7 cm, BD 8 cm, find DA. x 3.6 DA 4.67 x D x A B 3 x + 6 C Find the perimeter The triangles below are similar. Solve for x. x 6 9 x 9 24 x Given that ABC DEF find the following values: ABC _60 FDE _80 AB 9.6 EF 20 29
30 13. Write a similarity statement for each problem. Decide whether the triangles are similar and state why they are similar. a. Similarity Statement: ABC ~ EFD b. Similarity Statement: ABC ~ EFD Similarity Conjecture SSS Similarity Conjecture AA 14. A tree casts a shadow that is 30 long. If I walk along the shadow of the tree until my head is in a position where the end of my shadow exactly overlaps the end of the treetop s shadow. I am 20 feet from the end of tree and 10 feet from the end of the shadow. How tall is the tree if I am 6 6 tall? x 234 in 19.5 ft 30
31 15. Mrs. Holt is 5 6 tall and casts a 74 shadow. How tall is Mr. Sworsky, if at the same time, his shadow is one foot shorter? 66 x inches 16. Mary s eye is 144 cm above the ground and is standing 112 cm from a mirror on the ground. She looks into the mirror and sees the top of the tree. The mirror is 432 cm from a tree. How tall is the tree? (Round to the nearest tenth of a cm) 144 x x cm 17. A map uses the scale: 1 cm 8 kilometers. The Golden Gate Bridge is km long. a. How long would it be on the map? 1 x x 0.34 cm b. How many times larger is the real Golden Gate Bridge? (Hint: 1 km 1000 m, 1 m 100 cm) 8 km x 1000 m x 100 cm 1 km 1 m 800,000 times larger 31
32 18. A blueprint of a house has a living room that is 12 inches by 14.3 inches. The scale is 4 feet for every 3 inches. a. Find the real dimensions of the living room. 4 x 3 12, 4 x ft x 19 ft b. How many times bigger is the house than the blueprint? 4 ft x 12 in 48 : 3 1 ft 16 times bigger 19. Lines j, k, and GF are parallel. Find the missing values. A x x y x 7.2 m B 7.8 m 9.4 m C j y z 8.1 m E 12.7 m D k z G y z F 32
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