Aim: How are the side lengths related in 45-45-90 and 30-60-90 triangles?
Objectives: to apply the Pythagorean Theorem to determine the relationship between the side lengths
of 30-60-90 and 45-45-90 triangles.
Lesson Development: Whenever two side lengths are given in a right triangle, we can apply
Pythagorean Theorem to determine the third (missing) side. But there are two special right triangles
that we can determine the missing sides when only one side length is given.
Construct 3 squares with sides 1, 3 and 7, respectively. A diagonal is drawn for each square below,
which divides the square into two congruent isosceles right triangles. The hypotenuse of the
isosceles right triangle is always 2 times one of the legs or we say the legs and the hypotenuse are
in an extended ratio of 1:1: 2 . This is known as the 45  -45  -90  Triangle Theorem.
1
1
3
3
When dealing with 45-45-90 triangles, label legs
as x and hypotenuse as 2x as seen below.
7
7
When we need to solve for the lengths we just
compare:
In this case, the side facing the right angle is 15.
In the triangle on the left, it is 2x . so
2 x  12
. The two legs are 6 2 each.
12
x
6 2
2
There is another special right triangle whose side lengths are in an extended ratio.
A
2x
2x
B
C
2x
ABC is an equilateral triangle. We construct AD as an angle bisector. The two resulting triangles
will be congruent, which also makes AD a segment bisector. In other words, BD  CD by CPCTC.
The base has the height x and the height is
90 triangle Theorem
4 x 2  x 2  3x 2  3x . This is known as the 30-60-
We know 1  3  2 , so the sides facing 30 , 60 and 90 have an extended ratio of 1: 3 : 2 .
EX1: Find the value of each variable and express the answer in simplest radical form, if necessary.
ANS: x  4, y  4 3
ANS: x  10 3, y  15
ANS: x 
18
 6 3, y  12 3
3
We use the same idea here when we solve for the unknown sides. For example, in the middle
example, the side facing the right angle is 8, but in the theorem triangle, it is 2x. So 2x =8 gives us x
= 4. The side facing 30 is therefore 4, the side facing 60 is 3x  4 3 and the hypotenuse is 8.
EX2: Complete the table.
EX3: Complete the table
x
y
5
ANS: 4
AND: 5 2
4 2
2
ANS: 2
9
ANS: 9 2
ANS: 12 2
24
EX4: You are replacing the roof on the house
a) Find the values of x and y in the diagram.
shown, and you want to know the total surface
area of the roof. The roof has a 1-1 pitch on both
24
sides, which means that it slopes upward at a rate x  2  12 2
of 1 vertical unit for each horizontal unit.
y  12 2
b) Find the total surface area of the roof to the
nearest square foot.
A  35(12 2)  35(12 2)  1187.9
HW#5: P302 – 303: 1 – 16, 23, 25, 27.
HW#5 Solutions
1)
2)
a) 4
a) 2/3
b) 4
b) 2/3
3)
a) 5
a) 3
b) 5
b) 3
a) 3 2
a) 7
b) 3 2
b) 7
9)
a) 4 2
5
a)
2
2
a) 7
10)
a) ¼
11)
a) 5
12)
13)
a) 2 3
a) 5
b) 4
5
b)
2
b) 7
1
b)
4
b) 5
b) 6
14)
a) 13/2
15)
a)
16)
a) 3 3
4)
5)
6)
7)
8)
23)
x6 2
y  12
c) 4 2
2
c)
2
3
c) 10
c) 3 2
c) 6
c) 14
c) 8
c) 5
2
2
c) 14
3
c) 1/2
3
c) 10
3
c) 4 3
c) 10
b) 5 3
13
b)
3
2
b) 3
3
c) 13
c) 2 3
b) 9
25)
x 8 2
y4 6
c) 6 3
27)
OB  2
OC  2
OD  2 2
OE  4
EX1: Find the value of each variable and express the answer in simplest radical form, if necessary.
EX2: Complete the table.
EX3: Complete the table
x
y
5
2
4 2
9
24
EX4: You are replacing the roof on the house shown, and you want to know the total surface area of
the roof. The roof has a 1-1 pitch on both sides, which means that it slopes upward at a rate of 1
vertical unit for each horizontal unit.
a) Find the values of x and y in the diagram.
b) Find the total surface area of the roof to the nearest square foot.