Journal Chapters 7 & 8 Salvador Amaya 9-5

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

Examples Proportion 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: /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=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

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/ x=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 tall Is 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

Examples What is sinA in the following triangles? 16/20 or 4/5 6/13 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? 8/14 or 4/7 5/15 or 1/312/17

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 is Of 1/3. What is the perimeter of the New triangle? 6x1/3= 2 3x1/3= =5 cm 6 cm 3 cm

Examples The scale factor for the new triangle Is of 2. What is the perimeter of the New triangle? 16x2=32 20x2=40 9x2= =90 cm 20 cm 16 cm 9 cm

Examples The scale factor for the new triangle Is of ¼. What is the perimeter of The new triangle? 56/4=14 41/4= /4= =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 new Triangle? 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 cm

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

Examples What is tanC in the different triangles? 30/18 or 15/9 4/10 or 2/5 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 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 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= 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

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

_____(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.