Ratio – Comparison of two quantities measured in the same units The distance to Centralia is about 20 miles compared to about 100 miles to Vancouver. C:V = 1:5 Ratios in class: 1) girls to boys, 2) boys to girls, 3) girls to students _____ , ______ , ______

Proportion – Two or more ratios combined with an equal sign = means solve proportions – product of extremes equal to product of the means. a:b = c:d extremes

Extended Proportion 376/1 – 10, 12 – 42 even A AB : BC : AC = 7 : 5 : 4 2x + 1 x + 2 B C 2x – 5

2/16 O.E. What is the difference between a ratio and a rate? . Which terms are the means? Find the average (arithmetic mean) of 16 and 36.

Properties of Proportions Given: Cross Product Property Reciprocal Property May be rates as well as ratios. Interchange Means Property Nameless Property

Solving Problems involving Proportions The ratio of pounds of potatoes to gallons of soup is 2:5. If 125 pounds of potatoes are used, how large is the batch of soup? Verbal Model Ratio Given Constant Ratio Present Ratio pounds 2 pounds 125 pounds ; = gallons 5 gallons 312.5 gallons x gallons

If the plan of a house below is drawn to a scale of 1/8 inch to 1 foot what are the dimensions of the house? If the hallway is 3 feet wide how large should it appear on the plan? 8” 3” actual 3’ wide

The ratio of Gear A to B is the same as the ratio of B to C The ratio of Gear A to B is the same as the ratio of B to C. If Gear A has 36 teeth, Gear C has 16 teeth, how many teeth does Gear B have C B A Note: B is said to be the geometric mean of A and B

In the previous slide if each window is 6’ tall what is the approximate height of the building’s wall? 382/1 – 29, skip 24 Verbal Model Ratio Given Constant Ratio Present Ratio ; = blocks 5 blocks 20 blocks structure height 6 feet x feet 24 feet

2/21 OE The ratio of cement to lime to gravel in a cubic yard of concrete 5:1:32. If a batch of concrete contains 200 pounds of cement, what is the weight of the gravel? Given the proportion f:r = w:b, a) Write this proportion using fractions b) Re write the proportion using the cross product property c) Re write the proportion using the reciprocal property d) Re write the proportion using the means interchange property

L8.3 Similar Polygons

Definition – Two polygons are similar if their corresponding angles are congruent and the lengths of the corresponding sides are proportional. The symbol ~ is read “is similar to.” Not really necessary to be polygons x = 11 2/3 5 7 3

Similar Figures 5 7 2 7.5 6 8 Th. 8.1 – If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides.

Similar Figures Perimeter = 48 Perimeter = ? 4 6

Are circles similar? What are their corresponding parts?

390/1- 32 Are the two rectangles formed by this picture frame similar? 8” 10” 10” 12”

OE 2/22 List 5 ways to prove two triangles are congruent. Is ∆ABC ~ ∆XYZ, justify your answer. 3. Solve the proportion 9 15 6 12 8 Z Y X C B A

Lesson 8.5 Proving Triangles are Similar

AA postulate – if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. ΔABC ~ ΔXYZ C Y X A Z B

In addition to AA postulate Th. 8.2 SSS similarity – If corresponding sides of two triangles are proportional, then the 2 triangles are similar 9 12 10.5 6 7 8 9 11.5 11 7:8:6 = either 11:11.5:9 or 10.5:12:9 ?

Th. 8.3 SAS similarity – If an angle of one triangle is congruent to that of another triangle and the lengths of the sides including the angles are proportional, then the triangles are similar. 42o 16 24 42o 6 4 42o 6 8

SSS S SAS 4 AA 7 Q P 6 T R x

6’ 5’ 25’ x’ 397/ 7 – 12, 14 – 16 403/7 - 29, skip 21

OE 2/23 List the three ways to show two triangles are similar. Two congruent polygons are always, sometimes, never similar. What are the dimensions of the room drawn to scale below? Find the geometric mean of 4 and 9 2 – 1/8” Scale 1/16” = 1’-0” 1-1/4”

4. Given: ΔABC & ΔDEF are isosceles right triangles. Prove: ΔABC ~ ΔDEF A D C B F E 1. ΔABC & ΔDEF are isosceles right triangles. 2. AC  BC, DF  FE 3. C & F are right angles 4. C  F 5. 6. ΔABC ~ ΔDEF 1. Given 2. Def. of Isosceles triangles 3. Definition of right triangles 4. All right angles are congruent 5. Both ratios equal 1, substitution 6. SAS Similarity

13. Q Y 7 X Z 21 5 P R 70o 15 P 15. 12 Q 25O R 16 X 3 Y 4 Z

R Z 17. Given: ΔPQR & ΔXYZ are isosceles, R  Z Prove: ΔPQR ~ ΔXYZ X Y P Q Plan, show that sides are congruent and thus in proportion, show that P  X, then the triangles are similar by SAS postulate.

OE. 2/26 If two sides of a triangle are 27” and 38” long, what are the possible lengths of the third side? Find the value of x in each of the following. a) 15 7x + 2 b) 26 5x – 4 6x – 6

Lesson 8.6 Proportions in similar triangles

Th. 8.4 Triangle Proportionality Theorem – If a line parallel to one side of a triangle intersects the other two sides, then it divide the two sides proportionally. a c b d a c

Th. 8.5 Triangle Proportionality Converse– If a line divides two sides of a triangle proportionally, then it is parallel to the third side. a c b d

Caution – Th. 8.4 does not involve the parallel lines, whenever one of the parallel lines is involved you must use the definition to find solution. Definition – Corresponding sides are in proportion a c b d

Th. 8.6 If three parallel lines intersect two transversal, then they divide the transversals proportionally. Note – no triangle necessary a c b d

Application – Divide the following segment into 3 equal parts.

Th. 8.7 – 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 409/1 – 10, 11 – 25 odd, 30 – 35 a =25 c =10 b =x d =16

OE 2/27 If the measure of each interior angle of a regular polygon is 140o, how many sides does the polygon have? What is the relationship between a reflection line and the segment connecting corresponding points on the image and pre-image? Find the values of a, b, and c in the diagram below. ao 26o bo co 115o

Lesson 8.7 Dilations

A dilation is a non-rigid transformation, in which the image is similar to the pre-image. (projection)

0 < k < 1, the dilation is a reduction, k > 1, the dilation is an enlargement. A dilation is a non-rigid transformation, in which the image is similar to the pre-image. P’ 6 P C 4 C is center of dilation Ratio P’C : PC, is the scale factor. (image to pre-image), denoted as k k = 6 : 4 = 3:2 scale factor questions J The scale factor is the same as the ratio of corresponding sides. J’ If the arc length JP of the pre-image is 5.5” what is the arc length J’P’ of its image?

k = P’C : PC = 5 : 10 P STOP 5 P’ STOP 5 C Practice Sketch

Dilation, centered on the origin 415/1 – 4, 5 – 19 odd, 21 – 23, 25, 28 - 30 k = 2, (x,y)(2x,2y) P’(2,8) P(1,4) C(0,0) C’(0,0)

OE 2/28 A regular nonagon has a perimeter of 144”, what is the length of each side? Order the segments from shortest to longest. Find the value of x. R 38o 41o G K 12x - 9 7x + 6

OE 3/1 1. State which congruence postulate can be used to show ∆ABC  ∆CYA? What can you conclude about line l ? 3. List six ways to show a quadrilateral is a parallelogram. A B l

Test Review 1. Jim won 15 hands and lost 20 while playing blackjack. What is his ratio of winning hands to hands played? 2. Mary is digging a drainage ditch. After 2 hours she has dug 15 ft of ditch. At this rate how long will it take her to dig the entire 50 ft of ditch? Are the triangles similar, why? 2 6 34o 75o 4 34o 9 6 75o 3 78o

A Write all possible true proportions B C D E

A 53o B E Write the similarity statement for this diagram. C 53o D

The ratio of the sides of two similar buildings is 3:5, if the perimeter of the smaller building is 480, what is the perimeter of the larger? P 7 P’ 12 C

What is the scale factor? P = 46 P = ? D’ D 13 18

2/16 O.E. What is the difference between a ratio and a rate? . Which terms are the means? Find the average (arithmetic mean) of 16 and 36.

2/21 OE The ratio of cement to lime to gravel in a cubic yard of concrete 5:1:32. If a batch of concrete contains 200 pounds of cement, what is the weight of the gravel? Given the proportion f:r = w:b, a) Write this proportion using fractions b) Re write the proportion using the cross product property c) Re write the proportion using the reciprocal property d) Re write the proportion using the means interchange property

OE 2/23 List the three ways to show two triangles are similar. Two congruent polygons are always, sometimes, never similar. What are the dimensions of the room drawn to scale below? Find the geometric mean of 4 and 9 2 – 1/8” Scale 1/16” = 1’-0” 1-1/4”

OE. 2/26 If two sides of a triangle are 27” and 38” long, what are the possible lengths of the third side? Find the value of x in each of the following. a) 15 7x + 2 b) 26 5x – 4 6x – 6

OE 2/27 If the measure of each interior angle of a regular polygon is 140o, how many sides does the polygon have? What is the relationship between a reflection line and the segment connecting corresponding points on the image and pre-image? Find the values of a, b, and c in the diagram below. ao 26o bo co 115o

OE 2/28 A regular nonagon has a perimeter of 144”, what is the length of each side? Order the segments from shortest to longest. Find the value of x. R 38o 41o G K 12x - 9 7x + 6

OE 3/1 1. State which congruence postulate can be used to show ∆ABC  ∆CYA? What can you conclude about line l ? 3. List six ways to show a quadrilateral is a parallelogram. A B l