Colegio Herma. Maths Bilingual Department Isabel Martos Martínez. 2015

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1. Congruence

If shapes are identical in

shape and size then we

say they are congruent.

Congruent shapes can be mapped onto each other using

translations, rotations and reflections.

These triangles are congruent because

A

B

C

R

P

Q

AB = PQ, BC = QR,

and AC = PR.

A = P, B = Q,

and C = R.

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Congruent trianglesTwo triangles are congruent if they satisfy the following

conditions:

Side, side side (SSS)

1) The three sides of one triangle are

equal to the three sides of the other.

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Congruent triangles

2) Two sides and the included angle in one triangle are

equal to two sides and the included angle in the other.

Side, angle, side (SAS)

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Congruent triangles

Angle, angle, side (AAS)

3) Two angles and one side of one triangle are equal to

the corresponding two angles and side in the other.

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Congruent triangles

4) The hypotenuse and one side of one right-angled

triangle is equal to the hypotenuse and one side of

another right-angled triangle.

Right angle, hypotenuse, side (RHS)

If they are right-angled triangles:

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2. Similar shapesIf one shape is an enlargement

of the other then we say the

shapes are similar.

The angle sizes in two similar shapes are the same and their

corresponding side lengths are in the same ratio.

These triangles are similar because

A = P, B = Q,

and C = R.

A

B

C R

P

Q

PQ AB

= QR BC

= PR AC

A

B

C R

P

Q

A

B

C R

P

Q

A

B

C R

P

Q

A

B

C R

P

Q

A

B

C R

P

Q

A

B

C R

P

Q

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Similar shapes

triangles

A

B

C

A’

B’

C’

a’

a

b’

c’

c

b

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Similar shapesWhich of the following shapes are always similar?

Any two

squares?

Any two

rectangles?

Any two

isosceles

triangles?

Any two

equilateral

triangles?

Any two

circles?

Any two

cylinders?

Any two

cubes?

Similar polygons

Two polygons are similar if their corresponding angles are

equals and their corresponding sides are proportional.

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Finding the scale factor of an enlargement

We can find the scale factor for an enlargement by finding

the ratio between any two corresponding lengths.

Scale factor = length on enlargement

corresponding length on original

If a shape and its enlargement are drawn to scale, the the

two corresponding lengths can be found using a ruler.

Always make sure that the two lengths are written using the

same units before dividing them.

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The scale factor for the enlargement is 9/6 = 1.5

Finding the scale factor of an enlargement

The following rectangles are similar. What

is the scale factor for the enlargement?

6 cm 9 cm

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Finding the lengths of missing sides

The following shapes are similar. What is

the size of each missing side and angle?

6 cm

53°

a 5 cm

3 cm

4.8 cm 6 cm

b

The scale factor for the enlargement is 6/5 = 1.2

c

d

e

37°

37°

53°

3.6 cm

7.2 cm

4 cm

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Thales Theoreme

Given two straight lines, r and r´that cross at point O and that are

intersected by parallel segment 𝐴𝐴´ , 𝐵𝐵´ and 𝐶𝐶´ .

We can say that:

𝐴𝐵

𝐴´𝐵´=

𝐵𝐶

𝐵´𝐶´=

𝐴𝐶

𝐴´𝐶´

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THE USE OF THE THALES THEOREM

1. To divide a segment into equal parts:

We have segment AB and we want to divide it up into 5 equal

parts. We do not know the length of segment AB, so we use the

Thales’ Theorem to make the division.

1. Draw a line from one of the ends of the segment AB, in the

direction you want.

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2. As you have decided the line’s length, divide the line into 5

equal parts by using the compass.

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3. Join the last point with the end of the segment.

4. Draw parallel lines and the segment 𝐴𝐵 will be equally divided.

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So, you have divided up your line segment into 5 equal parts by

applying Thales theorem.

You can see the method in the following video:

How to divide a line into equal parts

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2. We use shadows and proportionality to find the length of

objects, height of buildings, persons, etc.

The rays of sunlight form a triangle with the ground and the line

of our object. The sunlight becomes the hypotenuse of a right-

angled triangle

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For example:

We want to know the height of a building:

1. We measure the real height of an object we have and the

length of its shadow projected on the ground.

2. We measure the length of building´s shadow projected on the

ground.

3. We can establish a proportion equality between the object, its

shadow, the building´s shadow, and the height we want to

know (x).

Therefore:

𝒔𝒉𝒂𝒅𝒐𝒘 𝒐𝒇 𝒕𝒉𝒆 𝒐𝒃𝒋𝒆𝒄𝒕

𝒓𝒆𝒂𝒍 𝒍𝒆𝒏𝒈𝒉𝒕 𝒐𝒇 𝒕𝒉𝒆 𝒐𝒃𝒋𝒆𝒄𝒕 =

𝒔𝒉𝒂𝒅𝒐𝒘 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒖𝒊𝒍𝒅𝒊𝒏𝒈

𝒓𝒆𝒂𝒍 𝒍𝒆𝒏𝒈𝒉𝒕 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒖𝒊𝒍𝒅𝒊𝒏𝒈 (𝒙)

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Using shadows to measure height

In ancient times, surveyors measured the height of tall

objects by using a stick and comparing the length of its

shadow to the length of the shadow of the tall object.

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If we closed these two figures and form two triangles, we

can guess they are similar, so their corresponding sides are

proportional.

a b

c

b á´

c´

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So, we can apply the proportinality rule for similar triangles:

b´ a´ b a

c´ c

Remember that:

The factor, k, is called the similarity ratio or scale factor

if k > 1 the shape is enlarged,

if k < 1 the shape is reduced.

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Exercise 1:

A particular map shows a scale of 1 : 500. What is the actual

distance if the map distance is 8 cm?

Exercise 2:

A particular map shows a scale of 1 cm : 5 km. What would the

map distance (in cm) be if the actual distance is 14 km?