Special Right Triangles

Special Right Triangles

SRT

Classifying Triangles

• Triangles are classified by their sides and angles

• There are some Right triangles that have special ratios of sides

• We call these Special Right Triangles

The 3-4-5 Triangle

This is the smallest Right Triangle that can be made with whole number sides, making the 3-4-5 uniquely special. It was used by ancient Egyptians and Babylonians long before Pythagoras was born.

The Equilateral Triangle

We already have learned that the Equilateral or Equiangular triangle has 3 congruent 60 degree angles and that all sides are congruent. Because of this, its easy to solve problems involving Equilateral Triangles.

Solving Equilaterals

Because all sides are congruent...

3x-1 = 11

3x = 12

x = 4

3(4)-1 = 11

Multiple Choice

What is the measure of BC?

• 12

• 6

• not enough information

• 10

Multiple Choice

What is the value of x?

• 16

• 60

• 8

• not enough information

Multiple Choice

Find x

• 3

• 4

• 2

• 3/2

• 4/3

The 45-45-90 or $1:1:\sqrt{2}$  

This is an Isosceles Right Triangle.
Because the two legs are congruent, the hypotenuses is always  $\sqrt{2}$   times the leg

Using PT to solve 

•  $1^2+1^2=c^2$  

•  $1+1=c^2$  

•  $2=c^2$  

•  $\sqrt{2}=c$  

Using Ratios

• b = 5 because its a leg of isosceles

• c =  $5\sqrt{2}$  

• ratio of sides is 5-5- $5\sqrt{2}$  

Multiple Select

What type of special right triangle is this?

• Isoceles right

• 30-60-90

• spatial

• 45-45-90

Multiple Choice

What is side c?

• hypotenuse

• da kine

• right angle

• leg

Multiple Choice

Find x.

• 90°
• 45°
• 30°
• 60°

Multiple Choice

In this 45-45-90 triangle, I have been given a leg, so to find the other leg I...

• Multiply that leg by 2

• Use the same length for the second leg

• Multiply that leg by √2

• Divide that leg by √2

Multiple Choice

In this 45-45-90 triangle, I have been given the length of a leg. How do I find the length of the hypotenuse?

• It is the same length as the given leg.

• Multiply that leg's length by √2.

• Multiply that leg's length by 2.

• Divide that leg's length by √2.

Multiple Choice

What is the measure of side c?

• 7

• $\frac{7}{\sqrt{2}}$

• $7\sqrt{2}$

• 14

Multiple Choice

What is the length of y in this picture?

• 45

• 5√2

• 90

• 5

Multiple Choice

What is the length of x in this 45-45-90 triangle?

• 4√2

• 4

• 8

• 4√3

Multiple Choice

Find x - the length of the hypotenuse of the triangle.

• 5
• 5√2
• 10
• 5√3

30-60-90 or
1- 2- $\sqrt{3}$  

30-60-90

• The short leg is opposite the 30 degree angle

• The hypotenuse is 2 times the Short Leg and is opposite the Right Angle

• The Long Leg is  $\sqrt{3}$  times the short leg and is opposite the 60 degree angle

• 2 is longer than  $\sqrt{3}$  !!!  $\sqrt{3}\approx1.73\ \ \ \ so\ \ \ \ 2>\sqrt{3}$  

Using ratios for 30-60-90

• Double the SL to get the Hyp

• Halve the Hyp to get the SL

• multiply the SL by  $\sqrt{3}$  to get LL

• divide the LL by  $\sqrt{3}$  to get SL

Multiple Choice

What is the measure of angle B?

• 90

• 30

• 60

• 45

Multiple Select

Which of these describe the Triangle...

• 45°-45°-90°

• 30°-60°-90°

• Right

• 1- 1- $\sqrt{2}$  

• 1- 2- $\sqrt{3}$  

Multiple Choice

What are x and y in this 30-60-90 triangle?

• x = 6 y = 3√3

• x = 3√2 y = 3

• x = 3√3 y = 6

• x = 6 y = 3√2

Multiple Choice

How long is c?

• 3

• 6

• 6√3

• 6√2

Multiple Choice

How long is e?

• 3

• 6

• 6√3

• 6√2

Multiple Choice

Solve for c

• 6

• 3

• 12

•  $6\sqrt{3}$ 

Multiple Choice

Find the value of y.

• 8

• 4

• 2√3

• 8√3

Multiple Choice

What are a and b in this 30-60-90 triangle?

• b=3.5√3 a = 7√3

• b = 7 a = 7√2

• b = 7 a = 14

• b = 7√3 a = 14√3

Working with Square roots

How would you solve this? lets call the leg

 $x^2$ = 2     ......take sq rt both sides
 $\sqrt{x^2}=\sqrt{2}$  

 $x=\sqrt{2}$   
this makes sense because to get back to the hypotenuse multiply leg by  $\sqrt{2}$  ..

 $\sqrt{2}\times\sqrt{2}=2$  .  

Solve Using Proportions

.....consider comparing the leg to the hyp.  again we will call the leg x...  the fraction on the right is the ratio for the 45-45-90

 $\frac{L}{Hyp}=\frac{x}{2}=\frac{1}{\sqrt{2}}$   

 $x=\frac{2}{\sqrt{2}}$  ...but there is a problem

Rationalizing the Denominator

we found x =  $\frac{2}{\sqrt{2}}$    but as a rule we dont leave radicals in the denominator.  To rationalize, we need to multiply by a fraction equal to 1 so that we change the way the answer looks, but not its value...

 $\frac{2}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\ \frac{2\sqrt{2}}{2}=\sqrt{2}$  
see the second fraction is =1         we find the Leg to be  $\sqrt{2}$  

Multiple Choice

What fraction would you multiply by to rationalize this  $\frac{9}{\sqrt{5}}$ 

•  $\frac{9}{9}$   

•  $\frac{\sqrt{5}}{5}$ 

•  $\frac{9}{\sqrt{5}}$ 

•  $\frac{\sqrt{5}}{\sqrt{5}}$ 

Multiple Choice

Simplify by rationalizing:  $\frac{9}{\sqrt{5}}$ 

•  $3\sqrt{5}$ 

•  $\frac{9\sqrt{5}}{5}$ 

•  $\frac{3\sqrt{5}}{5}$ 

•  $\frac{9}{25}$ 

Multiple Choice

Fully simplify....

Multiple Choice

Rationalize the denominator.

 $\frac{4}{\sqrt{5}}$  

•  $\frac{4\sqrt{5}}{5}$  

•  $\frac{\sqrt{5}}{4}$  

•  $\frac{\sqrt{6}}{3}$  

•  $\frac{5\sqrt{2}}{2}$  

Put it all together

• What type of SRT is it?

• What is the ratio for sides for that SRT?

• Use scale factor or proportions to solve for side

• rationalize if needed

• check answer for reasonableness

Multiple Choice

Find a.

• 2

• 4

• 2/√2

• 2√3

Multiple Choice

What is the length of y in this 45-45-90 triangle?

• 8√2
• 4√2
• 4
• 8

Multiple Choice

Find the value of y.

• 9

• 18√2

• 9√2

• (9√2)/2

Multiple Choice

Find the value of x.

• 18

• 18√3

• 36√3

• 12

Multiple Choice

Find the value of y.

• 11√2

• 11

• 22

• (11√2)/2

Multiple Choice

Find the value of y.

• 7

• 7√3

• 14√3

• (14√3)/3

Multiple Choice

Find the value of y.

• 8

• 4

• 2√3

• 8√3

Equilaterals & SRT

If you draw an altitude in an Equlateral triangle, it divides into two 30-60-90's

What is the height?

h =  $3\sqrt{3}$  

Square Diagonals & SRT

If you draw the diagonal of a square, it divides into two 45-45-90's
so the diagonal of any square is  $\sqrt{2}$   times the side lenght.

Multiple Choice

What is the altitude of this triangle?

• 12

• 6

• $6\sqrt{2}$

• $6\sqrt{3}$

Multiple Choice

A square has a side length of 8cm. What is the length of its diagonal?

• 16cm

• 10cm

• $8\sqrt{2}cm$

• $8\sqrt{3}cm$

Multiple Choice

Determine the value of x.

• x = 8√2

• x = 16√2

• x = 16

• x = 8√6

Multiple Choice

Determine the value of c.

• c = 6√3

• c = 12√3

• c = 18

• c = 18√2

Multiple Select

All of theses are 30-60-90 triangles where the SL of 1st (smallest) triangle is half the LL of next bigger triangle and so on..   Solve for a

• first LL is 3

• Second LL is  $3\sqrt{3}$  

• Third SL is  $3\sqrt{3}$  

• a = 9

Poll

Pau! How you feeling about all this stuff?

• All good

• SRT is good, rationalizing is annoying

• a little bit lost really

• totally confused

• what just happened?