Perimeter and Area Relationships

They really get around!

Learning Goals

By the end of today’s lesson I will:

• recognize the relationship between the area of similar triangles
• recognize the relationship between the perimeter of similar triangles

Triangle Review

In order for us to discuss triangles, we have to be able to identify the 6 different types of triangles.

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We will start by checking what you remember...

Hangman Q1

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___ ___ ___ ___ ___ ___ ___ ___ ___

• 2 sides of the same length
• 2 identical angles

Hangman Q2

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___ ___ ___ ___ ___

• The square in the corner means the two lines meet at 90^o
• 2 other angles are acute

Hangman Q3

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___ ___ ___ ___ ___

• all internal angles are less than 90^o
• the longest side is opposite the largest angle

Hangman Q4

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___ ___ ___ ___ ___ ___

• One angle is greater than 90^o
• The longest side is opposite the largest angle

Hangman Q5

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___ ___ ___ ___ ___ ___ ___

• No sides of equal length
• No identical angles
• (This one is acute)

Hangman Q6

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___ ___ ___ ___ ___ ___ ___ ___ ___

• 3 sides of the same length
• 3 identical angles
• All angles are 60^o

Similarity similarity similarity similarity similarity

If two triangles are similar, then they have the same shape.

This means that their angles will always be the same

Find the missing Information

If we have 2 similar triangles, we can find missing information:

3 cm

4 cm

5 cm

b

12 cm

a

Livin’ on the Edge

Their perimeters are similar as well:

3 cm

4 cm

5 cm

9 cm

12 cm

15 cm

You’re just like me! Only Different!

… and so are their areas:

3 cm

4 cm

5 cm

9 cm

12 cm

15 cm

What is the common link?

The lengths of our sides are all multiplied by 3. How does that relate to our perimeter and area?

3 cm

4 cm

5 cm

9 cm

12 cm

15 cm

• Our perimeter is multiplied by 3
• Our area is is multiplied by 9

Why 9?!?!

Our area being multiplied by 9 seems a little odd. Let's see why it works:

There are 9 of our original triangle that fits into our similar scaled up version

Original

Similar

Why 9?!?!

Let’s look at it mathematically using the area formula:

Original

Similar

Recap

If similar triangles have sides of 3 times the length of the original, then the area is 9 times the area of the original

In general, if the scale factor is x, the area is x² times the size.

Determine the length of the missing sides

5 cm

12 cm

13 cm

b

24 cm

a

d

c

39 cm

f

6 cm

e

Determine the perimeter of the similar triangles

3 cm

4 cm

5 cm

10 cm

2.5 cm

20 cm

Determine the area of the similar triangles

5 cm

12 cm

13 cm

10 cm

18 cm