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Similarity, Right Triangles and Trigonometry Journal Ch. 7 & 8

Similarity, Right Triangles and Trigonometry Journal Ch. 7 & 8. By: Mariana Beltranena 9-5. Ratios and Proportions. Ratio: is a comparison by division between two quantities.

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Similarity, Right Triangles and Trigonometry Journal Ch. 7 & 8

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  1. Similarity, Right Triangles and TrigonometryJournal Ch. 7 & 8 By: Mariana Beltranena 9-5

  2. Ratios and Proportions • Ratio: is a comparison by division between two quantities. • Proportion: is an equality between two equal ratios. A proportion can be solved by using the cross product properly and solving the resulting equation. • Their relationship: a proportion and ratio are related because a proportion is an equality between two equal ratios.

  3. examples • Ratio: 2/3, 6/9, 15/18 • Proportion: 6/10=12/20, 9/18=1/2, 15/18=20/24 • Solving proportions:

  4. Similar Polygons • When we say that two polygons are similar it means that both have the same shape but different sides. However, the sides in the similar polygons are all in the same ratio. The ratio of one side of the polygon to the corresponding side of the other polygon is called the scale factor.

  5. examples

  6. how to use similar triangles to perform an indirect measurement? • To use similar triangles to perform an indirect measurement you first create a similar triangle with known data, then measure the side that can be easily measured in the desired triangle and then solve for the unknown value that we want to know. • It is an important skill to know because it allows measures of objects that are to far or to big to be measured directly.

  7. Examples of indirect measurements 37m/9m= Htree/2 74 = 9 htree H tree=8.22m H? 2m 9m 37m

  8. Example 2 Find DF: AC/DF=BC/EF 64in/DF=24/105 24(DF)=64*105 DF= 280 or 23 ft 4 in D B A C F 2ft E 8ft 9 in

  9. Example 3 Find the height of the flag pole: 5ft/14ft 2 in= 5 ft 6 in/ h? (5)h= 70ft 12 in H= 15 ft 7 in is the height of the flag pole H? 5 ft 6 in 14 ft 2 in 5 ft

  10. using the scale factor to find the perimeter and area of a new similar figure • The ratio of perimeters = ratio of sides (scale factors) • Ratio of areas= (ratio of sides) ² Example: if the area of triangle ABC is 90. AB=4, ABC similar to FGH and FE=2. Find a) the area of FGH. • (4/2) ²= 90/ A2 • 4/1= 90/ A2 • 4A2= 90 • A2= 22.5 ft is the area of FGH

  11. Example 2 • If the ratio of the side of a quadrilateral, ABCD, to the side of quadrilateral FGHI is ¾; both polygons are similar. The perimeter of polygon ABCD is 36 ft, find a) the perimeter of FGHI b) the area of FGHI. • ¾= 36/P2; (3/4) ² --9/16= 72/A2 • 3P2/3= 144/3; 9 A2= 1,152 • P2= 48 ft perimeter of FGHI; A2= 128 ft area of FGHI

  12. example3 G B F A 10 14 5 C H E J D I If polygon ABCDE is similar to polygon FGHIJ Find the ratio of the perimeters Find the ratio of the areas Ratio of sides= 14/10= 7/5 Ratio of perimeters = 7/5 Ratio of areas (7/5)² = 49/ 25

  13. The three trigonometric ratios • Sin=Opposite/Hypotenuse • Cos=Adjacent side/Hypotenuse • Tan=Opposite side/Adjacent • What means to solve a triangle is to find all the sides and angles of a triangle.

  14. EXAMPLES y 24 Find x,y and angle a: Sin24= 12/x 12/(sin24)=x x= 29.5 Angle a= 66 degrees. Tan24= 12/y 12/(tan24)=y y=26.9 2) Using the calculator find each measure: Cos^-1(1/3)= 70.5 12m x 3) Find X: Cos38= x/18 18(cos38)=x X=14.1 <a 25 x 18 38

  15. Angle of elevation and angle of depression • Angle of elevation: angle between line of sight and horizontal where you see an object upward. • Angle of depression: angle between line of sight and horizontal when you see an object at a lower level. • Both are used the same way to get missing parts of angles or measurements.

  16. ANGLE OF DEPRESSION ANGLE OF ELEVATION

  17. 32 Tan32= 2/x 2/tan32 ???= 3.2 km 2 km ??? 32

  18. THE END!!!!

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