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

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.

examplesRatio: 2/3, 6/9, 15/18

Proportion: 6/10=12/20, 9/18=1/2, 15/18=20/24Solving proportions:

Similar PolygonsWhen 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.

examples

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.

Examples of indirect measurements

2m

9m

H?

37m

37m/9m= Htree/274 = 9 htreeH tree=8.22m

Example 2

2ftA

B

C8ft 9 in

E

D

F

Find DF:AC/DF=BC/EF64in/DF=24/10524(DF)=64*105DF= 280 or 23 ft 4 in

Example 3

14 ft 2 in

H?

5 ft

5 ft 6 in

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

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/ A24/1= 90/ A24A2= 90A2= 22.5 ft is the area of FGH

Example 2If 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/A23P2/3= 144/3; 9 A2= 1,152P2= 48 ft perimeter of FGHI; A2= 128 ft area of FGHI

example3A B

C

DE

10 5

FG

H

I

J

14

If polygon ABCDE is similar to polygon FGHIJa) Find the ratio of the perimetersb) Find the ratio of the areas

Ratio of sides= 14/10= 7/5Ratio of perimeters = 7/5Ratio of areas (7/5)² = 49/ 25

The three trigonometric ratios

Sin=Opposite/HypotenuseCos=Adjacent side/HypotenuseTan=Opposite side/AdjacentWhat means to solve a triangle is to find all the sides and angles of a triangle.

EXAMPLESy

x

24

12m

Find x,y and angle a:Sin24= 12/x12/(sin24)=xx= 29.5

Angle a= 66 degrees.

Tan24= 12/y12/(tan24)=yy=26.9

<a

2) Using the calculator find each measure:Cos^-1(1/3)= 70.5

x

25

18

3) Find X: Cos38= x/1818(cos38)=xX=14.1

38

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.

ANGLE OF DEPRESSION

ANGLE OF ELEVATION

32

32???

2 km

Tan32= 2/x2/tan32???= 3.2 km

THE END!!!!