JOURNAL CHAPTER 7 & 8

1. JOURNAL CHAPTER 7 & 8 BY: ANA JULIA ROGOZINSKI (YOLO)

2. RATIOS -A ratio is a comparison between one number to another number. In ratios you generally separate the numbers using a colon ( : ) between them or using a fraction. EXAMPLES: Alexandra has in her backpack 7 books, 13 crayons, 2 pencils, and 8 markers. 1.If you want to write the ratio of books to crayons, you would write it 7:13 2. If you want to write the ratio of pencils to markers, you would write it 2:8 3. If you want to write the ratio of pencils to crayons, you would write 2:13 4. The ratio of the side lengths of a quadrilateral is 2:4:5:7 and its perimeter is 36m. What is the length of the longest side? 2x +4x+5x+7x=36 18x=36 x=2 7(2) = 14

3. PROPORTIONS -A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.In a proportion a/b = c/d, a and d are the extremes and b and c are the means. EXAMPLES: 1.3:4 = 6:8 or 3/4 = 6/8 2.8:10 = 4:5 or 8/10 = 4/53. 25:15 = 5:3 or 25/15 = 5/34. 2:3 = 16:24 or 2/3 = 16/24

4. RATIOS/PROPORTIONS - A ratio and a proportion are related in the form that a ratio relates two quantities together using a fraction, and a proportion establishes a relationship between two quantities. Also they both show an equation with a ratio.- To solve a proportion you use cross products property. ( The product of the extremes is equal to the product of the means). If one term of a proportion is not known, cross multiplication can be used to find the value of that term.For example a/b = c/d, then ad=bc.1. x/2= 40/16 2. 7/y = 21/27 3. x²/18 = x/ 6 16x = 80 189 = 21y √6x² = √18x x=5 9=y x=3 or x=0 EXAMPLES:

5. Checking a proportion -To check if a proportion is equal you simplify both fractions until they are both equal, then if they are you have it correct meaning that they are equal if not you may have done something wrong. EXAMPLES: 1. 6/8 and 3/4, you reduce 6/8 to 3/4 2.15/25 and 5/3, you reduce 15/25 to 5/33. 8/10 and 4/5, you reduce 8/10 to 4/5 so both are equal.

6. SIMILAR POLYGONS - Similar figures don’t necessarily need to have the same size but they do need to have the same shape. Two polygons with corresponding angles congruent and their corresponding side lengths are proportional are SIMILAR POLYGONS. EXAMPLES: 3. 2. NO 1. YES YES

7. SCALE FACTOR - A Scale Factor describes how much the figure is enlarged or reduced. It’s used on each dimension in order to change one figure into a similar figure. As well it is a form of describing how the sides of a polygon re different. DILATION: transformation that changes the size of the figure but not its shape. EXAMPLES: 3. 1. 2.

8. Using Similar Triangles to perform an indirect measurement INDIRECT MEASUREMENT: any method that uses formulas, similar figures or proportions to measure an object.-To make an indirect measurement by using similar triangles you use the given measurements and information in order to find your answer. This is an important skill because there are many ways to use in real life, for example when you want to find the measurement of tall objects for example buildings, trees, etc.

9. EXAMPLES: 1. 2. 3.

10. Using the scale factor to find the perimeter and area -To find the scale factor for the perimeter and areas of similar figures you need to see the corresponding sides which are proportional and the corresponding angles are equal. The scale factor describes the difference between the side it also describes how much a figure is enlarged or reduced. PERIMETER: when you have the perimeter of both triangles then form fractions putting the smaller shape over the bigger shape and simplify. The ratio is the same as the ratio of their sides.AREA: Follow the same steps as in finding the perimeter but after you have made the fractions then simplify them all it can, and then you square them.

11. EXAMPLES: 6 3. 1. 4 8 6 6 4 4 10 6 4 Perimeter: 2/3Area:4²/9² 16 20 2. Perimeter: 14(4)=5624(24)=9656/96=7/12 Perimeter: 36/72 = 1/2 24 14 Area: if sides where 8 and 248/24= 1²/3² 4 Area:8/16=2²/4² 2 4 2

12. Three trigonometric ratios -A trigonometric ratio is the ratio a two sides of a triangle.- The three Trigonometric function are Sine, Cosine, and Tangent.SinA is the ratio of the length of the opposite side / hypotenuseCosA is the ratio of the adjacent / hypotenuse (between 0 and 1)TanA is ratio of the opposite / adjacent INppositeypotenuseOSdjacentypotenuseANppositedjacent SOHCAHTOA opposite hypotenuse adjacent

13. -They can be used to solve a right triangle because it helps you find the missing sides and angles of the triangle. -To solve a triangle means to find every angle and side of it. EXAMPLES: SinA: 6/10= 3/5CosA: 8/10 = 4/5Tan= 6/8 = 3/4Using the converse you know that A=37 and B=53 1. B 10 6 A 8 2. SinA: 5/13CosA:12/13TanA:5/12 13 3. SinB: 12/13CosB:5/13TanB:12/5 5 12

14. ANGLE OF ELEVATION AND DEPRESSION -An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line.-An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. - They are different in the fact that the angle of elevation is above the line and the angle of depression below the line.Theyare similar in the fact that both are angles formed with a horizontal line. Angle of depression Angle of elevation

15. EXAMPLES: 1. How is the angle that the arrow is pointing called? ANGLE OF ELEVATION 3. 25 5 25 Tan25=5/x5/tan25=xx=4.3 2. What would “x” be representing?The angle of elevation

16. THE END THE END THE END MR. YOLO