Review of Ratio and Proportion

Ratio- a comparison of two quantities. the ratio of p and q is

q

p or p :q or p to q

The ratio of to is _______________________

EX1/ Find the ratio of shaded boxes to unshaded .

EX2/ Find the ratio of the perimeter of the right triangle to the area of the right triangle.

Proportions- a statement that two ratios are equal

Is the following statement a proportion? (are they equal?)

70

30

35

15

Solve each problem. In Problems 1–3, solve for x. In Problems 4– 8, measurements are in centimeters.

1. x

25 =

4

5 2.

3

13 =

18

x 3.

16

x =

x

9

4. A recipe for 5 dozen cookies calls for 3

4 cups of shortening. How many cups of shortening are

needed to make 8 dozen cookies?

2

5

12

10

2

3

6

Corresponding angles are congruent

7’

5’

2’

4’

15

AC

ac

BC

bc

AB

ab

a b

c

A B

C

11.1 SIMILARITY

Two polygons are similar polygons if and only if 1) Corresponding angles are congruent and

2) Corresponding sides are proportional

Must be true for #1 and #2 for two shapes to be similar.

Which of the following are similar? Why?

10

12

6

10

5

15

3

10

2

A B D E

F 5 6

3 2

C The symbol for similar to is ~ 1. ABC~ DEF

Find the dimensions of the remaining sides.

hint: since they are similar, they are proportional

4. Find the perimeter of the larger pentagon if the two are similar.

Perimeter = _________

If one polygon is the image of another polygon under a dilation,

then the polygons are similar. 5. Draw a similar figure that is not congruent using the graph paper. hint: turn each square into a bigger square.

4

11.2 SIMILAR TRIANGLES ( Just for triangles ) Congruent triangles review: (eliminate the two that are not always true)

Side-Side-Side Side-Side-Angle Side-Angle-Angle

Angle-Angle-Angle Angle-Side-Angle Side-Angle-Side

Now for Similar Triangles:

SSS similarity Conjecture If the three sides of one triangle are proportional to the three sides of another triangle, then the two

triangles are similar.

AA similarity Conjecture If two (why not three?) angles of one triangle are congruent to two angles of another triangle, then

the triangles are similar. Think of it as a d ialation of one of the triangles.

SAS similarity Conjecture If two sides of one triangle are porportional to two sides of another triangle and the included angles

are congruent, then the two triangles are similar.

6

A

B

C E

ABC

D

~ Why?

A

B

G

F

EFGE

~ Why?

C 15’

10’

3’

2’

1. Is BAK ~ JOL? (yes/no),why = _________ 2. Is ACE ~ SPY? (yes/no), why = _________

3. 4.

5. Draw the two similar triangles from number 3 separately below.

8

11.3 INDIRECT MEASUREMENT WITH SIMILAR TRIANGLES.

tree

me

meofshadow

meofheight

treeofshadow

treeofheight

'5.2

'6

'11

h

What is the height of the tree?

1. A massive dog casts a shadow of 10 ft. A near by 40 ft build ing casts a 65 foot shadow.

Approximately how tall is the dog?

2. A 1 inch tall roach casts a 2 inch shadow. At the same time a cat’s shadow is 13 inches longer.

How tall is the cat?

3. A roach is attached to a string clipped to a head band of a 4.2 ft child standing straight up with

perfect posture. The roach is stretching the 5 ft string tight trying to get away. At the same time that

the roach is at the end of the small child 's shadow a cat, watching with amazement, is casting a 1.2 ft

shadow. How tall is the cat? Draw a picture of the situation.

10

A B

C D

11.4 CORRESPONDING PARTS OF SIMILAR TRIANGLES

PROPORTIONAL PARTS CONJECTURE: If two triangles are similar,

then the corresponding altitudes, corresponding medians, and

corresponding angle bisectors are proportional to the corresponding sides.

Draw the two sets of similar

triangles created by the medians

for the similar triangles below.

The angle bisector in a triangle d ivides the opposite side into two segments whose

lengths are in the same ratio as the lengths of the two sides forming the angle.

SIDEOPPOSITEBIG

sideoppositesmall

SIDEBIG

sidesmall

D

C

B

A

12

5

10 15

5

14

7

1

4

4

4

1 1

11.5 PROPORTIONS WITH AREA AND 11.6 VOLUME

AREA: If two similar polygons(or circles) have lengths of corresponding sides (or

radii) in the ratio of n

m, then their areas are in the ratio of

2

2

n

m.

Let's look at similar rectangles and similar triangles:

Ratio of the sides= Ratio of the sides=

Now try to find the ratio of the areas without the bases

Ratio of the areas = Ratio of the areas =

VOLUME: If two similar solids have the corresponding d imensions

in the ratio of n

m, then their volumes are in the ratio of

3

3

n

m.

1) Find the ratio of the sides 2) Find the ratio of the volumes(see how many little cubes you can fit in

the big cube)

Ratio of the sides=

Ratio of the volumes=

ratio of corresponding sides =𝑠𝑖𝑑𝑒

𝑠𝑖𝑑𝑒 ratio for area =

𝑠𝑖𝑑𝑒2

𝑠𝑖𝑑𝑒2 ratio for volume =

𝑠𝑖𝑑𝑒3

𝑠𝑖𝑑𝑒3

1. The ratio of two corresponding sides for two similar triangles is 5

3. What is the ratio of their areas?

2. The ratio of two corresponding sides for two similar pentagons is 7

4. What is the ratio of their areas?

3. The ratio of two corresponding sides for two similar cones is 5

3. What is the ratio of their volumes?

4. The ratio of two corresponding sides for two similar cylinder is 7

4. What is the ratio of their volumes?

5.

length

length

area

area

width

width

6. If two objects are similar and the ratio of their areas is 25/81, what is the ratio of their dimensions?

7. If two objects are similar and the ratio of their areas is 25/81, what is the ratio of their volumes?

6. The corresponding sides of two similar trapezoids have a ratio of 3:5. If the area of the smaller is 63 square inches,

what is the area of the larger? The work is below. Area = _________

sideingcorresponderl

sidesmaller

arg5

3

areaerl

areasmaller

arg25

9

area of the smaller

x

63

25

9

area of the larger

14

7. The dimensions of a small pyramid are one third the dimensions of a larger one.

a) What is the ratio of the volumes?

b) If the volume of the larger is 540 cm3, what is the volume of the smaller one? Volume = _________

16

A C

D E

B

A C

D E

B

P Q

11.7 PROPORTIONAL SEGMENTS BY PARALLEL LINES

______

DEAC Draw and label the two similar triangles formed by the parallel

line.

Two things from this:

1) If a line parallel to one side of a triangle passes through the other two sides, then it forms two

similar triangles.

2) Conversely, If a line cuts two sides of a triangle forming two similar triangles, then it is parallel to a

side.

Draw and label the three similar triangles formed below. PQDEAC___