Activator • Solve the proportion: 4/ (x+2) = 16/(x + 5) • Simplify: 5weeks/30days; 85cm/.5m
UNIT E.Q • How can I use all what I am learning on polygon similarity to solve real life problems. • Which careers use similarity most? • Is similarity shown and used only in geometry?
Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2.Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6.Attend to precision. • 7. Look for and make use of structure. • 8.Look for and express regularity in repeated reasoning.
Essential Question: 1. How do we use proportions to solve problems? 2.How do we use properties of proportions to solve real- life problems, such as using the scale of a map?
Computing Ratios • Ratio of a to b : • Ex: Simplify the ratio of 6 to 8. Simplify the ratio of 12 to 4.
Simplifying Ratios • Simplify the ratios.
The perimeter of rectangle ABCD is 60 cm. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. • Hint: draw a rectangle ABCD.
Using Extended Ratios • The measure of the angles in ∆JKL are in the extended ratio of 1:2:3. Find the measures of the angles. 2x 3x x
Using Ratios • The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths. A D 3 L B C L F E 8
Properties of proportions • Cross product property then ad = bc. • Reciprocal property
Given that the track team won 8 meets and lost 2, find the ratios. • What is the ratio of wins to loses? • What is the ratio of losses to wins? • What is the ratio of wins to the total number of track meets?
Simplify the ratio. • 3 ft to 12 in • 60 cm to 1 m • 350g to 1 kg • 6 meters to 9 meters
Additional Properties of Proportions • If , then • If , then
Using Properties of Proportions • Tell whether the statement is true.
In the diagram . Find the length of BD. A 30 16 B C x 10 D E
In the diagram • Solve for DE. A 5 2 D B 3 E C
Geometric mean • The geometric mean of two positive numbers a and b is the positive number x such that • Find the geometric mean of 8 and 18.
Geometric Mean • Find the geometric mean of 5 and 20. • The geometric mean of x and 5 is 15. Find the value of x.
Different perspective of Geometric mean • The geometric mean of ‘a’ and ‘b’ is √ab • Therefore geometric mean of 4 and 9 is 6, since √(4)(9) = √36 = 6.
Geometric mean • Find the geometric mean of the two numbers. • 3 and 27 √(3)(27) = √81 = 9 • 4 and 16 √(4)(16) = √64 = 8 • 5 and 15 √(5)(15) = √75 = 5√3
Ex. 3: Using a geometric mean • PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x. Slide #26
Solution: The geometric mean of 210 and 420 is 210√2, or about 297mm. 210 x Write proportion = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 X = √210 ∙ 210 ∙ 2 X = 210√2 Cross product property Simplify Factor Simplify Slide #27
Using proportions in real life Slide #28 In general when solving word problems that involve proportions, there is more than one correct way to set up the proportion.
Ex. 4: Solving a proportion Width of Titanic Length of Titanic = Width of model Length of model LABELS: Width of Titanic = x Width of model ship = 11.25 in Length of Titanic = 882.75 feet Length of model ship = 107.5 in. Slide #29 MODEL BUILDING. A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it?
Reasoning: Write the proportion. Substitute. Multiply each side by 11.25. Use a calculator. Width of Titanic Length of Titanic = Width of model Length of model x feet 882.75 feet = 11.25 in. 107.5 in. 11.25(882.75) x = 107.5 in. x ≈ 92.4 feet So, the Titanic was about 92.4 feet wide. Slide #30
Note: Slide #31 Notice that the proportion in this Example contains measurements that are not in the same units. When writing a proportion in unlike units, the numerators should have the same units and the denominators should have the same units. The inches (units) cross out when you cross multiply.
A model truck is 13.5 inches long and 7.5 inches wide. The original truck was 12 feet long. How wide was it?
Identifying Similar Polygons • When there is a correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons.
Similar polygons • If ABCD ~ EFGH, then G H C D F B A E
Similar polygons • Given ABCD ~ EFGH, solve for x. G H C D 6 x B F 2 4 A E 2x = 24 x = 12
Is ABC ~ DEF? Explain. B D 6 E 13 12 5 7 A C F 10 ABC is not similar to DEF since corresponding sides are not proportional. ? ? yes no
Similar polygons Given ABCD ~ EFGH, solve for the variables. G H C D 5 x B F 10 2 y 6 A E
If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. Ex: Scale factor of this triangle is 1:2 9 4.5 3 6
Quadrilateral JKLM is similar to PQRS. Find the value of z. R K L S 15 Q z 6 P J M 10 15z = 60 z = 4
Theorem • If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. • If KLMN ~ PQRS, then
Given ABC ~ DEF, find the scale factor of ABC to DEF and find the perimeter of each polygon. E P = 8 + 12 + 20 = 40 B P = 4 + 6 + 10 = 20 12 20 10 6 A C D F 8 4 CORRESPONDING SIDES 4 : 8 1 : 2
In the diagram, ∆BTW ~ ∆ETC. • Write the statement of proportionality. • Find m<TEC. • Find ET and BE. T 34° E C 3 20 79° B W 12
Postulate 25 • Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
Similar Triangles • Given the triangles are similar. Find the value of the variable. )) m )) ) 6 8 11m = 48 ) 11
Similar Triangles • Given the triangles are similar. Find the value of the variable. Left side of sm Δ Base of sm Δ Left side of lg Δ Base of lg Δ = 6 5 > 2 6h = 40 > h
∆ABC ≈ ∆DBE. A 5 D y 9 x B C E 8 4