Unit 6:G e o m e t r i c a lProportionality

In this lesson you will learn about: Proportional segments Similar figures Tales theorem Maps and scales

Vocabulary you need in this lesson:You are going to use some vocabulary related to this lessonand the problems you will work with. The same words appe-ar in the crossword, in the word scramble and in the wordsearch.

Some theory The proportional ratio of two segments is the quotient between their

lengths. Segments AB and CD keep a proportional relationship with segments EF and

GH if:

Similar polygons have the same shape but different size.The lengths of their sides satisfy the relationship:

Their corresponding angles are equal: A = A’, B = B’ , C = C’ , D = D’

d d’

a c a’ c’b b’

Similar triangles: two triangles are similar if:3 angles of one triangle are the same as 3 angles of the other, or3 pairs of corresponding sides are in the same ratio (proportional), orAn angle of one triangle is the same as the angle of the other triangle and thesides containing these angles are in the same ratio.

Tales theorem: if two secant straight lines are cut by two , parallel lines, theratio of any two segments of one of the secant is equal to the ratio of thecorresponding segments of the other straight line.

A scale model is a representation or copy of an object that is larger orsmaller than the actual size of the object

'´'' d

d

c

c

b

b

a

a

GH

EF

CD

AB

Now it´s time to practiseExercises

1) Are two triangles similar with these sides?

2) Complete the measurements of the sides in these triangles to make them similar:

3 cm 6 cm

2 cm 4 cm 10 cm

3)What is the height of the tree?

4) Draw a segment with a length of 5 cm and divide it into 7 parts.

5) Are these polygons similar? What is the ratio? Explain your answer

6) A square has an area of 4 cm2 . What is the area of another square with a side:a) double? b) half?

7) Find similar triangles among these right-angled ones:

cmccmbcmacmccmbcma 10',18',12'.5,9,6

63º 41º 27º 45º 49º

47º

A C D EB F

8) There are four sets of similar triangles below. Can you work out the lengths of thesides marked with a question mark?

Problems

1) The length of a road on a map with a scale of 1:500 000 is 8 cm. What is its reallength?

2) The distance between Sevilla and Madrid is about 540 km. What is this distance on amap with a scale of 1:150 000.

3) Calculate the distance betweenTribecca and City Hall.

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14) See the map from Minnesota below. Suppose the scale it is using is 1:325000. Calculatethe following distances:

Mankato-Silver Bay

Austin-Fergus Falls

Minneapolis-Ely

15) The map below shows the routes taken by ships across the English Channel. Supposethe scale is 1 : 1 600 000

It takes 1.5 hours to travel from Dover to Calaix.Calculate the lengths of the routes shown on the map, to the nearest km.How long will each crossing take?Curiosity

Now we are going to study about a very important example of proportionality. Leonar-do da Vinci was was an Italian polymath, scientist, mathematician, engineer, inventor,anatomist, painter, sculptor, architect, botanist, musician and writer. Leonardo hasoften been described as the archetype of the Renaissance man, a man whose unquen-chable curiosity was equaled only by his powers of invention. He is widely considered tobe one of the greatest painters of all time and perhaps the most diversely talentedperson ever to have lived. According to art historian Helen Gardner, the scope anddepth of his interests were without precedent and "his mind and personality seem to ussuperhuman, the man himself mysterious and remote".

Leonardo da Vinci works in architecture and he also draws thehuman body. One of Leonardo's drawings is the Vitruvian Man. It isbased on a model of ideal proportions. The drawing shows a square in-scribed inside a circle. There is a man with two pairs of arms and legs,which touch both the circumference of the circle and the vertices ofthe square.

The conclusion can be that the length of a man's arm span isequal to the height of the man. In other words the ratio of the Vitru-vian Man's arm span to his height equals 1.

Question: Is the ratio of our arm span to our height really equal to 1?

Height Arm span

VITRUVIAN MAN

Some interesting websites!

If you want to practise more with proportionality you can go to:

http://gphillymath.org/IMP1Supp/Shadows.pdf (pages 1, 2, 3, and 4)

If you want to read more about similar triangles you can take a lookat:http://www.punchard.com/classes/geom/units/unit1-5.pdf

To read a bit more and see more examples about similarity go to:

http://www.mathsisfun.com/geometry/similar.html

To practice a little with similar figures after reading and examiningsome examples click here:

http://www.math.com/school/subject1/lessons/S1U2L4GL.html

If you want to read more about measuring distances on a map, go to:

http://geography.about.com/cs/maps/a/mapscale.htm

In this lesson you have read about a very important figure in science, artand a lot of different fields, Leonardo da Vinci. If you want to learnmore about him click here:http://en.wikipedia.org/wiki/Leonardo_da_Vinci%27s_personal_life