Warm Up
 Find x. Leave answer as simplified radical
ANSWER: 27 5
Special Triangles
Find the missing side lengths
• Can we use Pythagorean
Theorem?
30º
• Can we use similar
Triangles?
10
60º
• Any other way we know of
to find the missing side
lengths?
Before we begin…
Lets go over some vocabulary needed
ALTITUDE
HYPOTENUSE
LEG
The perpendicular height
from one side of a triangle
to the opposite vertex
The longest side of a right
triangle (the side across
from the right angle)
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
1
So the two bottom legs
must be congruent
2
1
2
We know all sides have a
length of 2. So if that side is
split into 2 congruent pieces
each piece must be 1.
3.What is the length of the third side (the
altitude)? Leave answer as simplified
radical.
1
1
Pythagorean Theorem:
1 + x2 = 4
x2 = 3
x= 3
X
12 + x 2 = 22
2
2
Quick Review:
How do we simplify radicals?
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.
120
12
6 2 5
3
2
10
2
so 120  2 30
Quick Review:
How do rationalize the denominator?
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.
18
3
∙
3
3
=
18 3
=6
3
3
Answer question 4 on your own or in
your pair
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.
Each 30°- 60°- 90° right triangle formed by drawing altitude
Side Length of
Equilateral Triangle
2 (first)
1 (second)
4
6
Hypotenuse
Length
Shorter Leg Length Longer Leg Length
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.
Each 30°- 60°- 90° right triangle formed by drawing altitude
Side Length of
Equilateral Triangle
Hypotenuse
Length
Shorter Leg Length Longer Leg Length
2 (first)
2
1
1 (second)
1
1/2
4
4
2
6
6
3
3
1
2
2
3
3
3
3
6. What is true about the lengths of
the sides of any 30°-60°-90° right
triangle?
30º
2x
x 3
60º
x
Foldable!
Once you have made your foldable
complete the table for question 7
Part 2: The baseball Diamond
A baseball diamond is, geometrically
speaking, a square turned sideways.
Each side of the diamond measures 90
feet. (See the diagram to the right.) A
player is trying to slide into home base,
but the ball is all the way at second base.
Assuming that the second baseman and
catcher are standing in the center of
second base and home, respectively, we
can calculate how far the second
baseman has to throw the ball to get it
to the catcher.
In your pairs answer question 8
9) Without moving from his position, the
catcher reaches out and tags the runner out
before he gets to home base. The catcher
then throws the ball back to a satisfied
pitcher, who at the time happens to be
standing at the exact center of the baseball
diamond.
a. What do we know about both of the legs
of this new triangle?
b. How can we solve for the legs?
c. Find the distance the catcher has to throw
to get it back to the pitcher. Leave
answers as simplified radicals.
P
In each 45°- 45°- 90° right triangle
Leg Length Other Leg
Length
Hypotenuse Length
90 ft.
45 2 ft.
11. What is true about the lengths of
the sides of any 45°-45°-90° right
triangle?
45º
x
x 2
45º
x
45°-45°-90°
triangle
hypotenuse
length
one leg length
other leg length

#1

#2
4 2 π 2
4
4
π
π

#3
11
11 2
2
11 2
2

#4
1
2
2

#5
#6

#7
6
7
6 3 5
3
3
3 10
2
3
3 10
2
2
3
2

7
7

#8
12 2
5
12
5
12
5
What are all the ways we NOW know
to find sides in a triangle




Pythagorean Theorem
Similar Triangles
30-60-90 triangles
45-45-90 triangles