Journal Chapter 7 & 8

Marcos Vielman 9-5

Ratios and Proportions

A ratio compares two numbers by division.

A proportion is an equation starting that two ratios are equal.

They are related because without ratios you can’t do proportions because a proportion needs two ratios that are equal.

You check if a proportion is equal by the Cross Products Property.

Examples

3-(-1)/4-(-2)= 4/6= 2\3

4-3/1-(-1)= ½

-2-2/-2-2= 0

x/2 = 40/16 = 16x = 80 = x= 5

7/y = 7/9 = 9y = 49 = x= 5.4

y/3 = 27/y = y2 = 81 = y= 9

Similar polygons and scale factor

For polygons to be similar means that if and only if their corresponding angles are congruent and their corresponding side lengths are proportional.

A scale factor describes how much the figure is enlarged or reduced.

Examples

D

A B

C

H

E

G

F

8

10

12

10.8

5.4

4

5

6ABCD ≈ EFGH<A ≅ <E<B ≅ <F<C ≅ <G<D ≅ <H

Rectangle ABCDRectangle A’B’C’DA(0,0) (0× 3/2, 0×3/2) A’(0,0)B(0,4)(0x 3/2, 4x 3/2) B’(0,6)C(3,4)(3x 3/2, 4x 3/2)C’(4.5,6)D(3,0)(3x 3/2, 0x 3/2)D’(4.5,o)Scale factor enlargement of a picture

Similar triangles with indirect measurement

You can use similar triangles to do indirect measurement in the way that if you draw an altitude from the vertex at the right angle of a right triangle, you form 3 similar right triangles.

You can use those 3 similar right triangles to find the indirect measurement.

This is an important skill because you can use it to find measurements of things you can’t measure by hand and your work can be faster and more efficient.

Examples

You can use the shadow of a tree measure it and find the indirect measure.

You can use the reflection of and object and how far you are from it to find the indirect measurement.

Also to find the indirect measurements you have to measure how tall you are.

Scale factor, perimeter and area

You find the scale factor by multiplying the vertices by how much you reduce or enlarge the image of what you are working with.

For the perimeter and the area of similar figures you can use proportions to find them.

Examples

The ratio of the side lengths of a quadrilateral are 2:3:5:7 its perimeter is 85 ft.

The ratio of the side lengths of a quadrilateral is 2:4:5:7 and its perimeter is 36m.

The ratio of the side lengths of an isosceles triangle is 4:4:7 and its perimeter is 52.5cm.

Trigonometric functions

SinA= opposite side/hypotenuse

CosA= adjacent/hypotenuse

TanA= opposite/adjacent

This functions can be used to solve a right triangle by finding the sides and the angles.

To solve a right triangle means to find all the angles and sides of the triangle.

Examples

a

A

B

C

c

b

SinA = a/cSinB = b/cCosA = b/cCosB = a/cTanA = a/b TanB = b/a

S

R

T

13

12

5sinR = 12/13 ≈ 0.92cosR = 5/13 ≈ 0.38tanS = 5/12 ≈ 0.42

Angle of elevation and depression

Angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line.

Angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line.

They are used to find out how much distance between objects that are elevated or are in the ground and seeing into elevated objects.

Examples angle of elevation

Examples angle of depression

THE END