Geometry Journal Chapter 7 & 8

1. Geometry Journal Chapter 7 & 8 By: Jaime Rich

2. What is a ratio? A comparison of two numbers by division. What is a proportion? An equation stating that two ratios are equal. How to solve them and how to check if they are equal: • You solve proportions by cross products or cross multiplication. • You check if they are equal if the product of the cross multiplication are equal to • each other. How are they related? Proportions are related to ratios because proportions are made up of ratios.

3. Ratios Proportions Solving Proportions a:b a:b = c:d a = c a = c b d b d 5:10 5:10 = 1:2 a/b a/b = c/d ad = bc 5/10 = 1/2 5/10 a to b a to b = c to d 5 to 10 5 to 10 = 1 to 2

4. Two polygons are similar when: Their corresponding angles are congruent and their corresponding side lengths are proportional. A similarity ratio is the ration of the lengths o f the corresponding sides of two similar polygons. Scale Factor: Shows how much a figure is enlarged or reduced.

5. 35 35 16 8 90 55 55 90 3 6 50 12 18 80 80 4 50 50 50 6

6. Indirect Measurement: Any method that uses formulas, similar figures, and / or proportions to measure and object. It is an important skill because: You can measure high, or big objects without physically measuring it with a ruler or Measuring tape. You can know distances from one place to another, and you can do many other things for planning for example to know where to build your house besides a tree pretending that tree may fall (you measure the height of the tree with indirect measurement and build the house away from it).

7. 2/6 = 6/h 2h = 36 H = 36/2 h H = 18 ft. 6 ft 2 ft 6 ft

8. 3/4.5 = 30/h 3h = 135 h = 135/3 h = 45 h 4.5 ft 3 ft 30 ft

9. 50/100 = 2.5/h 50h = 250 h = 250/50 h = 50 ft 100 ft h 2.5 ft 50 ft

10. How to use scale factor to find: Perimeter: Perimeter of small figure Area of small figure x x = = Area of larger figure Perimeter of larger figure y y Area: Then square root x/y and square that answer. -----> Theorem: If the similarity ratio of two similar figures is a/b, then the ration of their perimeters is a/b, and the ratio of their areas is a/b squared.

11. PERIMETER 22 / 66 = 2 / 6 = 1/3 30 15 10 The scale is 1 to 3 5 21 6/9 = 1/1.5 The scale is 1 to 1.5 7 3 1 6.5/26 = 1/4 2.5 2 1 The scale is 1 to 4 10 3 12 4.5 1.5 4 3

12. Area 17.5/157.5 = √(1/9) = (1/3)squared 15 30 10 5 21 1/2.25 = √(1/2.25) = (1/1.5)squared 7 3 1 2.5 2 1 10 3 12 4.5 1.5 4 1.5/24 = √(1/16) = (1/4)squared 3

13. Three Trigonometric Ratios: Sine: Ratio of the length of the leg opposite the angle to the length of the hypotenuse. Cosine: Ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. Tangent: Ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

14. To solve a right triangle: You can use the Pythagorean theorem OR use sin, cos, or tan to solve it which is much faster and easier. You can also use the inverse trigonometric functions (sin-1, cos-1, and tan-1). To solve a right triangle means: To find the measurements of sides and angles of right triangles.

15. SIN / COS / TAN Ex 1 R SinR = 12/13 = 0.92 CosR = 5/13 = 0.38 TanR = 5/12 = 0.42 13 5 S T 12 Ex 2 Cos(76) 0.2419218956 Ex 3 Sin-1R = 67 Cos-1R = 68 Tan-1R = 23

16. Angle of Depression: Angle formed by a horizontal line and a line of sight to a point below the line. Angle of Elevation: Angle formed by a horizontal line and a line of sight to a point below the line.

17. Angle of Depression Angle of Elevation

18. < of depression < of elevation < of depression < of elevation