a b denominators cannot be zero must have the same units must be simplified 12cm 6ft 24oz 14ft 440yd m 18in 2lb 6yd 2mi

Size: px

Start display at page:

Download "a b denominators cannot be zero must have the same units must be simplified 12cm 6ft 24oz 14ft 440yd m 18in 2lb 6yd 2mi"

Felicia Morris
6 years ago
Views:

1 Ratio of a to b a b a:b Simplifying Ratios: Converting: denominators cannot be zero must have the same units must be simplified 1 m = 100 cm 12 in = 1 ft 16 oz= 1 lb 3 ft = 1 yd 5, 280 ft = 1 mi 1,760 yd = 1 mi 12cm 6ft 24oz 14ft 440yd m 18in 2lb 6yd 2mi 6. The area of a rectangle is 108 cm If the measures of the angles in a triangle The ratio of the width to the length is 3:4. have the ratio of 4:5:6, classify the triangle Find the length and the width. as right, obtuse or acute.

2 Proportion: equation that equates two ratios Properties: a. Cross Products b. Reciprocal Property a c a c If =, then If =, then b d b d c. Interchange Property a c a If =, then b d c =. Practice: Complete each statement. 8. If 6 5, then 6 x y x = =. 9. If =, then =. x y y a = b c d x If =, then xy =. 4 y 9 x If =, then =. 2 y 9 Decide whether the statement is True or False. 12. If x 8, then y 3. y = 3 x = If x 8, then 3 y. y = 3 x = If x = 8, then x = 3. y 3 8 y 15. If x = 8, then x = y. y Solve for x. 16. x = = = x + 3 x 3 2x 7 x 3

3 Solve for the variable. O 19. MN:MO is 3:4 20. SU:UT = PR:RQ M S x 9 x N U 36 T P 5 R 12 Q Use the diagram and the given information to find the unknown length. 21. Given AB AE, find BC. BC = ED 22. Given AB = AE BC ED, find BC.. Geometric Mean: The geometric mean of two positive numbers a, b is the positive number x such that: 2 = so x = ab Geometric mean: x = ab 23. Find the geometric mean of 3 and Find the geometric mean of 6 and is the geometric mean of 4 and what number?

4 The perimeter and the ratio of the length to the width of a rectangle are given. Find the length and width of the rectangle. Draw a picture. 1. Perimeter: 132 cm 2. Perimeter: 280 ft l : w = 7:4 l : w = ll:9 The measures of the angles of a triangle are in the extended ratio given. Find the measures of the angles of the triangle. 3. 2:5:5 4. 3:7:10 Solve the proportion = 6. = 7. 8 y x x m + 5 = m x 9. y 2 2 y 3 = = 10. x z 2 z 2 = + 4 Find the geometric mean of the two numbers. 11) 2 and 8 12) 7 and 14 13) 10 and 12

5 In Exercises 18-19, the ratio of two side lengths for the triangle is given. Solve for the variable. 14) AB : AC is 2 : 1. 15) AB : AC is 15 : 8. Complete the statement: 7 = x 10? =. 10 y 7? ? = =. x y 24? 16. If, then 17. If, Then In the diagram, AB AG AB AG = and =. CD FE AC AF Find the unknown length. 18. Find AB 19. Find GF In the diagram, PQ = WV, QR = VU and PW = QV QR VU RS UT QV ST Find the unknown length. 20. Find UT. 21. Find QV.

6 Name: Block: Date: 1.) Simplify the ratio 1200 cm : 1.8 m. 2.) The perimeter of a rectangle is 528 millimeters. The ratio of the length to the width is 8:3. Find the length and the width. 3.) Solve 1 3 =. s ) Find the geometric mean of 42 and ) The extended ratio of the angles of a triangle is 5 : 12 : 13. Find the angle measures of the triangle. 6.) The area of a rectangle is 720 square inches. If the ratio of the length to width is 5 : 4, find the perimeter of the rectangle.

7 #7-8 Use the diagram and the given information to find the unknown length. 7.) Given XW WV XY =, find WV. 8.) YZ Given SR RQ = ST, find TU. TU #9-12 Decide whether the statement is true or false. 9.) x s y t If =, then =. 10.) y t x s x s x t If =, then =. y t s y 11.) x 6 x 4 If =, then = ) If a b and c d in the figure 13.) Given the statement: below, what is the value of x? Prime numbers are always odd. A 143 o A valid counterexample would be that the B 133 o number ---- C 57 o A 39 is odd D 47 o B 38 is even C 17 is a prime number D 2 is a prime number 14.) Consider the following true 15.) If M is the midpoint of CD, MD = 2x + 5, statement: p ~q and CM = 4x 5, what is x? Which of the following is a valid conclusion? A q p A -5 B q ~p B 0 C ~q ~p C 5 D ~p ~ q D 15

Similar Polygons: SYMBOL for SIMILAR: Corresponding angles are Corresponding sides are Writing Similarity Statements: ABC Corresponding < s: Proportional Sides: (statement of

8 Similar Polygons: SYMBOL for SIMILAR: Corresponding angles are Corresponding sides are Writing Similarity Statements: ABC Corresponding < s: Proportional Sides: (statement of proportionality) A BC XYZ = = = If 2 polygons are, then the ratio of the lengths of 2 corresponding sides is called the. What is the scale factor of ABC to XYZ? Practice: 1) 2)

9 3) = = You Try: 1.) If polygon LMNO~HIJK, complete the proportions and congruence statements. Hint: Draw a diagram!! a) M b) K c) N MN HK HI IJ HK d) = e) = f) = IJ J K L M MN 2.) In the diagram, polygon ABCD ~ GHIJ. 8 A B G y H D 11 x x 11 C a. Find the scale factor of polygon b. Find the scale factor of polygon ABCD to polygon GHIJ. GHIJ to polygon ABCD. 5.5 J 8 I c. Find the values of x and y. d. Find the perimeter of each polygon. e. Find the ratio to the perimeter of ABCD to perimeter of GHIJ.

10 Altitude/Height of a Triangle: the segment from a base to the opposite vertex. In similar triangles, the altitudes will also be in proportion. If ABC DEF, find the length of the altitude in ABC. If 2 polygons are, then the ratio of their perimeters is equal to the ratios of their. If 2 polygons are, then the ratio of any two corresponding lengths in the polygons is equal to their. 3.) The ratio of one side of ABC to the corresponding side of similar DEF is 3:5. The perimeter of DEF is 48in. What is the perimeter of ABC?

12 List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality (an extended ratio of corresponding side lengths). 1. ABC DFE 2. WXYZ ~ MNOP A AB = = W WX = = = Determine whether the polygons are similar *look at your definition of similar*. If they are, write a similarity statement and find the scale factor. If not, state why not In the diagram, WXYZ MNOP. 5. Find the scale factor of WXYZ to MNOP. 6. Find the values of x, y, and z. 7. Find the perimeter of WXYZ. 8. Find the perimeter of MNOP. 9. Find the ratio of the perimeter of MNOP to the perimeter of WXYZ.

13 The two triangles are similar. Find the values of the variables. Remember, all corresponding lengths in similar figures are in proportion. (Including altitudes!) Multiple Choice The ratio of one side of ABC to the corresponding side of a similar DEF is 4:3. The perimeter of DEF is 24 inches. What is the perimeter of ABC? A. 18 inches B. 24 inches C. 32 inches In the diagram, XYZ MNP. 13. Find the scale factor of XYZ to MNP. 14. Find the unknown side lengths of both triangles. 15. Find the length of the altitude (the segment perpendicular to ZY ) shown in XYZ.

14 Name: Block: Date: Simplify the ratio feet 30in 2. 6 in. 1 mi cm 3m Use properties of proportions. 4. If 2 c =, then 3 3 h 2 =??. 5. If 2 x y = 5, then 2? x =? 6. Find the geometric mean of 12 and Find the geometric mean of 13 and 6. Draw a diagram 8. The measures of the interior angles of a triangle are in the extended ratio of 1:3:5. Find the measure of each angle. Label the largest and smallest. 9. Solve for x. x 8 = x 1 7

15 10. AB AC DE =, find EF. Be careful!!!! DF A D 8 10 B E 2 C F Solve for x given that AC:BC is 7:2. A 3x + 6 B x C 12. In December 2000, the exchange rate of Mexican pesos to US dollars was 7.56 to 1. You paid 240 pesos for a jacket. What was the price of the jacket in US dollars? Round to the nearest cent. (May use calculator for this.) 13. The perimeter of a rectangular field is 56 yd. The ratio of its length to its width is 6:4. What is the length and width of the field? (Hint: Draw a picture) 14. JLK ~ XZY. (a) Find the scale factor of A to B and then (b) Find the missing length.?

16 In the diagram ABCD ~ WXYZ. Show all work, round each answer to the nearest tenth and write your answer on the line provided Find the scale factor of ABCD to WXYZ. A u D s 3.6 W 14 Z 16. Find the scale factor of WXYZ to ABCD. 17. Find the value of s, t, and u. s =, t =, u = B 3 C 8.4 t 115 o 18. Find the measure of angle C. X 7 Y 19. Find the ratio of the perimeter of ABCD to the perimeter of WXYZ.

17 AA Triangle Similarity Theorem If two angles of one triangle are to two angles of another triangle, then the triangles are. SAS Triangle Similarity Theorem If two triangles have angles are pairs of proportional side lengths and the, then the triangles are similar. SSS Triangle Similarity Theorem If two triangles have pairs of side lengths proportional, then the triangles are.

18 Practice: Determine whether the triangles are similar. If they are, state what postulate or theorem you used and write a similarity statement. (remember, order matters!) If not explain why. 1.) 2.) 3.) 4.) Show that the two triangles are similar. Write a similarity statement and the postulate or theorem used to prove the similarity. 5.) ABE and ACD 6.) SVR and UVT

19 7.) SRT and PNQ 8.) HGJ and HFK 9.) A flagpole casts a shadow that is 50 feet long. At the same time, a woman standing nearby who is five feet four inches tall casts a shadow that is 40 inches long. How tall is the flagpole to the nearest foot? 10.) A larger cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 ft wide and 25 feet long. The old court was similar in shape, but only 16 ft wide. a) Find the scale factor of the new court to the old court b) Find the perimeters of the new court and the old court

20 11.) Find the value of x that makes ABC ~ DEF. Hint: set up and solve proportions. 12.)

21 Name: Block: Date: Use the diagram to complete the statement. 1. ABC ~ 2. BA AC CB = = (which thm/post proves this?) = = y = 6. x = Determine whether the triangles are similar. If they are, state the theorem or postulate use and write a similarity statement. If they are not similar, state Not Similar thm/post: ~ thm/post: ~ thm/post: ~ thm/post: ~

11. 12. 13. *You might find it helpful to draw LQN and MPN separately *You might find it helpful to draw ACE and BCD separately thm/post: thm/post: thm/post: ~ ~ ~ 14.

22 *You might find it helpful to draw LQN and MPN separately *You might find it helpful to draw ACE and BCD separately thm/post: thm/post: thm/post: ~ ~ ~ 14. Is either JKL or RST similar to ABC? 15. Ruby is standing in her back yard and she decides to estimate the height of a tree. She stands so that the tip of her shadow coincides with the tip of the tree s shadow, as shown. Ruby is 66 inches tall. The distance from the tree to Ruby is 95 feet and the distance between the tip of the shadows and Ruby is 7 feet. a. What postulate or theorem can you use to show that the triangles in the diagram are similar? b. About how tall is the tree, to the nearest foot?

23 Name: Geometry HW Similar Triangles: Applications 1. A tree 24 feet tall casts a shadow 12 feet long. Brad is 6 feet tall. How long is Brad's shadow? (draw a diagram and solve) 2. Triangles EFG and QRS are similar. The length of the sides of EFG are 144, 128, and 112. The length of the smallest side of QRS is 280, what is the length of the longest side of QRS? 3. A 40-foot flagpole casts a 25-foot shadow. Find the shadow cast by a nearby building 200 feet tall.

24 Name: Geometry 4. A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post? 160 cm 90 cm 360 cm 5. A tower casts a shadow 7 m long. A vertical stick casts a shadow 0.6 m long. If the stick is 1.2 m high, how high is the tower? 6. A tree with a height of 4m casts a shadow 15 m long on the ground. How high is another tree that casts a shadow which is 20 m long? (draw a diagram and solve)

25 Name: Geometry 7. Triangles CDE and NOP are similar. The perimeter of smaller triangle CDE is 133. The lengths of two corresponding sides on the triangles are 53 and 212. What is the perimeter of NOP? 8. Luisa began walking up a hill at a spot where the elevation is 0.9 km. After she walked 3 km, she saw a sign giving the elevation as 0.95 km. How far will she have walked when she reaches and elevation of 1.1 km? Draw a diagram. 9. The foot of a ladder is 1.2 m from a fence that is 1.8 m high. The ladder touches the fence and rests against a building that is 1.8 m behind the fence. Draw a diagram, and determine the height on the building reached by the top of the ladder. 10. Jermaine is painting a mountain scene from a 3 3 inch by 6 inch postcard. If he wants the enlargement to 5 be similar to the original, which of these dimensions should he choose for his canvas? A. C. 4 2 inch by 8 inch 5 B. 5 inch by 7 inch 7 1 inch by 12 inch 5 D. 8 inch by 10 inch 12. A rectangular soccer field is 120 yards long by 64 yards wide. What dimensions would a scale drawing of the soccer field be if the field were drawn using the scale ¼ inch = 10 feet? 13. In the diagram shown, a metal support brace is added to stabilize a metal triangle. The brace is parallel to the 18-foot base of the triangle and divides the left side into a 4-foot and a 6-foot section. What is the length of the metal support brace?

26 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two intersected sides proportionally. Note this theorem doesn t involve the parallel sides TU andus Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Examples: 1.) In the diagram, QS UT, RS = 4, ST = 6, and QU = 9. What is RQ? T ****Stop: is there another way of looking at this?**** In fact, we must apply this process when we are asked to find lengths of the middle pieces (the parallel segments) 2.) Find QS and UT. T

27 We can use the converse of the Triangle Proportionality Theorem to determine if segments are parallel or not. 3.) Determine whether PS QR. Theorem 6.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. Example: 4.) Find the length of AB. Theorem 6.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Example: 5.) Find the length of AB.

28 Mixed Practice #5-9: Use the diagram to find the value of each variable. 5.) 6.) 7.) Hey! You re asked to find the parallel segments lengths! You better pull apart those triangles! 8.) 9.) x

29 #10-13: Determine each length using the diagram to the right. Draw the triangles if you need to! 10.) AG 11.) FC 12.) ED 13.) AE

30 Use the figure to complete the proportion. 1. GC CF = DB 4. AE CD = GE 2. AF FC = BD 5. FG AG = FB 3. CD FB = GD 6. GD GE = AE Use the given information to determine whether BD AE Determine the length of each segment. 11. BC 12. FC 13. GB 14. CD

31 In Exercises 15 18, find the value of x Are the lines parallel? How do we know? Find the value of the variable 19. x 20. m 21. a 22. Maps On the map below, 51st Street and 52nd Street are parallel. Charlie walks from point A to point B and then from point B to point C. You walk directly from point A to point C. a. Using the proportionality theorems, how many more feet did Charlie walk than you?

32 RECAP Similarity Shortcuts for Triangles AA SSS SAS There are 3 similarity shortcuts ways you can tell triangles are similar. AA Similarity Conjecture SSS Similarity Conjecture If two angles of one triangle are congruent to If the three sides of one triangle are proportional two angles of another triangle, then the to the three sides of another triangle. Then the triangles are similar. two triangles are similar. Here are some ways that you can find similar triangles: Angles are marked congruent Vertical Angles are congruent Alternate Interior Angles are congruent SAS Triangle Similarity Conjecture Two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar. Corresponding Angles are congruent Shared Angle Reflexive Property Richard Sudo Monday, December 7, :54:39 PM CT 00:19:e3:4a:d2:21

33 Examples: 1. We know that these triangles are similar by Find x. 2. We know that these triangles are similar by Find x. 3 Richard Sudo Monday, December 7, :54:39 PM CT 00:19:e3:4a:d2:21

34 MATH-G Similar Triangles SOL Practice [Exam ID:2E63V2 Don't forget about congruent triangles! 1 Which triangle below is not congruent to the other three triangles? A C B D 2 What value of x makes ΔDEF ΔJLK? F x = 6.0 G x = 5.3 H x = 4.1 J x = 9.4 Think about order. Try out each option. 3 If triangle XYZ is similar to triangle XLM, then A XL : LM = YZ : XZ B XL : LY = XZ : MZ C XM : XZ = XL : XY D XM : XZ = XY : XL 4 Given: Therefore F LM MN G LM LP LN H LP LN J LM = PQ QL = MN QP = LQ LM = NP MQ

35 What does it take for the triangles to be similar? Does that work for every side provided? 5 Triangle LMN is similar to triangle PQR. Which of the following sets of side lengths could be those of triangle LMN? A 2 in., 3in., 4 in. B 6 km, 7 km, 8 km C 8 ft, 15 ft, 17 ft D 9 m, 12 m, 15 m What reason do you have to say they are similar? Think about how you would prove it. 6 Which drawing contains a pair of similar triangles? F H G J There are many ways to do this. I would try to pick the easiest method. But thats just me. *Make sure to read the whole question.* 7 F 11 G 24 H 12 J 22

36 Name: Block: Date: 1. Simplify the ratio 2. Simplify the ratio 3. Find the geometric 2 lb 10 ft : 3 yd 24 lb mean of 6 and 24. Solve for x = x 1 2x x = 3x The perimeter of a rectangle is 56 inches. The ratio of the length to width is 6:1. Find the length and width. Length = Width= 7. The area of a rectangle is 525 square cm. The ratio of the length to width is 7:3. Find the length and width. Length = Width= 8. The ratio of the measures of the angles of a triangle is 7:14:15. Find the measure of each angle.

37 9.) Given the diagram, identify the following terms. a.) In ABD, identify the vertex angle. b.) In ABD, identify the legs., d.) In ABD, identify the legs., 10.) Complete the sentence with always, sometimes, or never. a.) An isosceles triangle is a right triangle. b.) An isosceles triangle is an equilateral triangle. c.) An obtuse triangle is an isosceles triangle. 11.) x = y = 12.) x = ; y = 13.) x = 14.) x = 15.) x =

38 Solve for each variable. Show all work. 16.) x = 17.) x = ; y = 18.) x = ; y = 19.) x = 20.) What is the perimeter of the triangle? 21) In isosceles FOB, O is the vertex angle. If m F = (7x 3) and m B= (3x+ 17), find the measure of each angle. (draw a picture!) m F = m O= m B=

39 22 Solve the proportion: 23. Find CA. 2 5 = x 1 3x In the diagram, DEFG~PQRS. (a) Find the scale factor of DEFG to PQRS. (b) Find the value of x. (c) Find the value of y. (d) Find the value of z. 25. Given: LMN ~ PQR. Find the values of the variables. M (a) a = b = Q (b) What is the scale factor of LMN ~ PQR? (c) What is the perimeter of PQR? 12 R a b L N 16 P (d) What is the scale factor of the perimeter of PQR to the perimeter of LMN?

40 #26-31: State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity statement. If not, write not similar #32-33 Given that the two triangles are similar, find the value of x x x + 5 y x

34. Use the diagram to find the missing value of h. 35. A boy knows that his height is 6 feet. At the time of day when his shadow is 4 feet, a tree s shadow is 24 feet. What is the height of the tree?

41 34. Use the diagram to find the missing value of h. 35. A boy knows that his height is 6 feet. At the time of day when his shadow is 4 feet, a tree s shadow is 24 feet. What is the height of the tree? 36. The shadow of the flagpole is CE. CE is 80 ft. If the shadow of a 12 ft street-lamp is 30 ft, how tall is the flagpole? D A C B E 37. When a 2-meter stick (standing vertical) casts a shadow 3 meters long, a 30 meter tree also casts a shadow. How long is the tree s shadow? Draw and label a sketch. Then write a proportion and solve.

42 38. Find the height of each figure in the picture. Each one is an enlargement of the preceding one. The smallest figure s height is already given. #4 #3 (a) Figure #1 (b) Figure #2 #2 #1 (c) Figure #3 (d) Figure # (e) What is the scale factor of figure 4 to the original figure? # Find the value of x. SHOW ALL WORK x x 5

Chapter 6: Similarity

Name: Chapter 6: Similarity Guided Notes Geometry Fall Semester CH. 6 Guided Notes, page 2 6.1 Ratios, Proportions, and the Geometric Mean Term Definition Example ratio of a to b equivalent ratios proportion