Journal Chapters 7 & 8 Salvador Amaya9-5

Ratio

•Comparison of 2 numbers written a:b, a/b, or a to b.

Examples

The ratio of adults to kids in this family is 2:1

The ratio of red fruits to the rest is of 2:4

The ratio of cars to motorcycles is 3:6

Proportions

•Compares ratios saying they are equal.

Examples

Proportion

1/4= 3/12

Examples

Proportion

2/3= 4/6

ExamplesProportion

1:2= 3:6

How to Solve a proportion

•If there are 2 variables▫Cross multiply▫Square root both sides▫Use + and – root to solve for x

•If there is only one variable▫Cross multiply▫Divide

Example

•8/y=12/4•Cross multiply: 32/12y•Divide: ---------• 12 •8/3=y

Example

•9/x+2=x+2/4•Cross multiply: 36=(x+2)2•Square root: +-6=x+2•Solve: 6-2=4• -6-2=-8•x= 4, -8

Example

•3/7=x/12•Cross multiply: 36=7x•Divide: -------• 7•36/7=x

How to check if a proportion is equal

•Cross multiply and check if the 2 products are equal

Example

•Check if this proportion is equal•2/4=5/20•Cross multiply: 2x20=40, 4x5=20•The products are not equal, so the

proportion is not equal.

Example

•Check if this proportion is equal•4/7=16/21•Cross multiply: 4x21=84, 7x16=84•The products are equal, so the proportion

is equal.

Example

•Check if this proportion is equal•3/9=12/13•Cross multiply: 3x13=39, 9x12=108•The products are not equal, so the

proportion is not equal.

Similar polygons

•They have congruent corresponding angles and their corresponding sides are proportional.

Examples

Examples

Examples

Scale Factor

•It tells you how much the picture is enlarged or reduced.

Using similar triangles for indirect measurement•To measure something that is too tall to

measure it with a ruler or a meter stick, you can use the sun rays. You stand so the sun makes a shadow and you measure your height and your shadow measure. Then you measure the shadow of the object you want to measure to make a proportion and find the height of the object.

Example

•He wants to find the height of the tree. He is 1.7 m. tall and his shadow is of 2 m. The tree’s shadow is of 5 m. How tall is a tree?

Make the Proportion

•Height of boy/shadow of boy=height of tree/shadow of tree

•1.7/2=x/5•2x=8.5•x=4.25•The tree is 4.25 m. tall.

Example

•He wants to find out the height of the house. He is 1.8 m. tall and his shadow is of 2.1 m. The house’s shadow is of 4.6 m.

•How tall is the house?

Make the Proportion

•Height of boy/shadow of boy=height of house/shadow of house

•1.8/2.1=x/4.6•2.1x=8.28•x=3.9•The house is 3.9 m. tall.

Example

•He wants to find out the height of the dinosaur. He is 1.6 m. tall and his shadow is of 1.8 m. The dinosaur’s shadow

Is of 30 m. How tallIs the dinosaur?

Make the Proportion

•Height of boy/shadow of boy=height of dinosaur/shadow of dinosaur

•1.6/1.8=x/30•1.8x=48•x=26.7•The dinosaur is 26.7 m. tall.

Trigonometric Ratios

•Sine (Sin): opposite side/hypotenuse•It can never be more than 1

a

Examples

•What is sinA in the following triangles?

a

2016

16/20 or 4/5

A13

6

6/13

a

7

35

7/35 or 1/5

Trigonometric Ratios

•Cosine (Cos): adjacent side/hypotenuse•It can never be more than 1

Examples

•What is cosB of the different triangles?

b 155

B

14

8

8/14 or 4/7

a

17

12

5/15 or 1/3 12/17

Using scale factor to find perimeter

•Since you are given the lengths of the triangle and then a fraction that tells you how much it is enlarged or reduced, you multiply the lengths times that fraction to get the new sides. You then add all the sides to get the perimeter.

Examples

•The scale factor for the new triangle isOf 1/3. What is the perimeter of the New triangle?•6x1/3= 2•3x1/3=1•2+2+1=5 cm

6 cm 6 cm

3 cm

Examples

•The scale factor for the new triangleIs of 2. What is the perimeter of theNew triangle?•16x2=32•20x2=40•9x2=18•32+40+18=90 cm

20 cm16 cm

9 cm

Examples

•The scale factor for the new triangleIs of ¼. What is the perimeter ofThe new triangle? •56/4=14•41/4=10.25•13/4=3.25•14+10.25+3.25=27.5 cm

56 cm.

41 cm.

13 cm.

Using scale factor to find area

•Since you are given the lengths of the triangle and then a fraction that tells you how much it is enlarged or reduced, you multiply the lengths times that fraction to get the new sides. You then use the triangle area formula to get the area of the new triangle.

Examples

•The scale factor for the new triangle Is of ½. What is the area of the newTriangle?•18/2=9•3/2=1.5•½(1.5)x9•3/4x9=6.75 cm2

20 cm

3

18 cm

Examples

•The scale factor for the new triangle is Of 3. What is the area of the new Triangle?•12x3=36•5x3=15•½(15)x36•7.5x36=270 cm2

5

12

Examples

•The scale factor for the new triangle is of 1/5. What is the area of the new triangle?

•22/5=4.4•20/5=4•½(4)x4.4•2x4.4=8.8 cm2

22

20

Trigonometric Ratios

•Tangent (Tan): opposite side/adjacent side•It can be more than 1

Examples

•What is tanC in the different triangles?

c30

18

30/18 or 15/9

C4

10

4/10 or 2/5

c17

40

40/17

Solving a right triangle

•To solve a right triangle refers to find out all of its sides and all of its angles.

How to solve a right triangle using trigonometric ratios•To find the length of a side:•Write a ratio that can be written with the

info you have•Leave the side you want to find alone•Solve

Examples

•What is the length of side AB?•We have the hypotenuse and the opposite

side of angle 41°, so we’ll use sine•Sin41=AB/18•18sin41=AB•11.80=AB

41

18 cm

a b

Examples

•What is the length of side OP?•We have the adjacent side and the

opposite side of angle 56°, so we’ll use tangent

•Tan56=OP/26•26Tan56=OP•38.55=OP

56 26

p

o

Examples

•What is the length of side UV?•We have the adjacent side and the

hypotenuse of angle 35°, so we’ll use cosine.

•Cos35=23/UV•UV=23/Cos35•UV=28.08

3523

v

u

Angle of Elevation

•Angle formed by a horizontal line and a line above the horizontal line.

•It is congruent to the angle of depression.

Examples

Examples

Examples

Angle of Depression

•Angle formed by a horizontal line and a line below the horizontal line.

•It is congruent to the angle of elevation.

Examples

Examples

Examples

_____(0-10 pts) Describe a ratio. Describe a proportion. How are they related? Describe how to solve a proportion. Describe how to check if a proportion is equal. Give 3 examples of each. _____(0-10 pts) Describe what it means for two polygons to be similar. What is a scale factor? Give at least 3 examples of each. _____(0-10 pts) Describe how to use similar triangles to perform an indirect measurement. Why is this an important skill? Give at least 3 examples. _____(0-10 pts) Describe how to use the scale factor to find the perimeter and area of a new similar figure. Give 3 examples of each, 3 for perimeter, 3 for area. _____(0-10 pts.) Describe the three trigonometric ratios. Explain how they can be used to solve a right triangle. What does it mean to solve a triangle? Give at least 3 examples of each. _____(0-10 pts.) Compare an angle of elevation with an angle of depression. How are each used? Give at least 3 examples of each.