Find x. Leave answer as simplified radical

Warm Up

ANSWER: 27 5

Special Right Triangles30 – 60 – 90

Find the missing side lengths

60º

30º

10

• Can we use Pythagorean Theorem?

• Can we use similar Triangles?

• Any other way we know of to find the missing side lengths?

Lets go over some vocabulary needed

Before we begin…

ALTITUDE

The perpendicular height from one side of a triangle to the opposite vertex

HYPOTENUSE

The longest side of a right triangle (the side across from the right angle)

LEG

The two sides that connect to the right angle in a right triangle.

Discovering Special Triangles

1. Adam, a construction manager in a nearby town, needs to check the uniformity of Yield signs around the state and is checking the heights (altitudes) of the Yield signs in your locale. Adam knows that all yield signs have the shape of an equilateral triangle. Why is it sufficient for him to check just the heights (altitudes) of the signs to verify uniformity?

Because all equilateral triangles are similar so one measurement will be sufficient.

Discovering Special Triangles

2. A Yield sign from a street near your home is pictured to the right. It has the shape of an equilateral triangle with a side length of 2 feet. If the altitude of the triangular sign is drawn, you split the Yield sign in half vertically, creating two 30°-60°-90° right triangles, as shown to the right. For now, we’ll focus on the right triangle on the right side. (We could just as easily focus on the right triangle on the left; we just need to pick one.) We know that the hypotenuse is 2 ft., that information is given to us. The shorter leg has length 1 ft. Why?

2 2

Congruent due to HL

2 2

So the two bottom legs must be congruent 2 2

We know all sides have a length of 2. So if that side issplit into 2 congruent pieceseach piece must be 1.

1 1

3.What is the length of the third side (the altitude)? Leave answer as simplified radical.

X

2 2

1 1

Pythagorean Theorem:

12 + x2 = 22

1 + x2 = 4

x2 = 3

x = 3

Break down the radicand (the number inside the radical) into perfect squares. Anything that is a perfect square will come out of the radical everything else stays inside the radical.

Quick Review: How do we simplify radicals?

120

12 10

5 26 2

3 2 so 120 2 30

We can never leave a radical in the denominator. Multiply the numerator and denominator by the radical on the bottom. This will get rid of the radical on the denominator, then simplify.

Quick Review: How do you rationalize the

denominator?

=

Answer question 4 on your own or in your pair

Side Length of Equilateral Triangle

Each 30°- 60°- 90° right triangle formed by drawing altitude

Hypotenuse Length

Shorter Leg Length

Longer Leg Length

2 (first)      

1 (second)      

4      

6      

5) Now that we have found the altitudes of both equilateral triangles, we look for patterns in the data. Fill in the first two rows of the chart below, and write down any observations you make. Then fill in the third and fourth rows.

Side Length of Equilateral Triangle

Each 30°- 60°- 90° right triangle formed by drawing altitude

Hypotenuse Length

Shorter Leg Length

Longer Leg Length

2 (first)  2 1   

1 (second)  1  1/2  

4 4  2   

6 6   3  

5) Now that we have found the altitudes of both equilateral triangles, we look for patterns in the data. Fill in the first two rows of the chart below, and write down any observations you make. Then fill in the third and fourth rows.

31

32

2 3

3 3

6. What is true about the lengths of the sides of any 30°-60°-90° right triangle?

2x

x

x

30º

60º

Foldable!

Once you have made your foldable complete the table for question 7