Section 7 – 3
Special Right Triangles
Objectives:
To use the properties of 45-45-90 triangles
To use the properties of 30-60-90 triangles
45-45-90 Triangles
Solve for y in terms of x.
45-45-90 Triangle Theorem
In a 45-45-90 triangle, both legs are
congruent and the length of the hypotenuse
is 𝟐 times the length of a leg.
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 = 𝟐 ∙ 𝒍𝒆𝒈
𝒍𝒆𝒈 =
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝟐
Example 1
Finding the Length of
the Hypotenuse
Find the value of each variable.
A)
B)
C) Find the length of the hypotenuse of a 4545-90 triangle with legs of length 𝟓 𝟑
D) Find the length of the hypotenuse of a 4545-90 triangle with legs of length 𝟓 𝟔
Example 2 Finding the Length of a Leg
Find the value of x.
A)
B)
C) Find the length of a leg of a 45-45-90
triangle with a hypotenuse of length 10.
D) Find the length of a leg of a 45-45-90
triangle with a hypotenuse of length 22.
Example 3
Real-World Connection
A) A square garden has sides 100 feet long.
You want to build a brick path along a diagonal
of the square. How long will the path be?
Round your answer to the nearest foot.
B) The distance from one corner to the
opposite corner of a square playground is 96
feet. To the nearest foot, how long is each side
of the playground?
C) You are designing dinnerware. What is the
length of a side of the smallest square plate on
which a 20-cm chopstick can fit along a
diagonals without any overhang?
Textbook Page 369; # 1 – 11 Odd
Section 7 – 3
Continued…
Objectives:
To use the properties of 30-60-90 triangles
30-60-90 Triangles
Label the sides of the triangle as
HYPOTENUSE, LONG LEG, or SHORT LEG.
30-60-90 Triangle Theorem
In a 30-60-90 triangle, the length of the
hypotenuse is twice the length of the
shorter leg. The length of the longer leg is
𝟑times the length of the shorter leg.
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 = 𝟐 ∙ 𝒔𝒉𝒐𝒓𝒕 𝒍𝒆𝒈
𝒍𝒐𝒏𝒈 𝒍𝒆𝒈 = 𝟑 ∙ 𝒔𝒉𝒐𝒓𝒕 𝒍𝒆𝒈
Example 4
Finding the Lengths of
the Legs
Find the value of each variable. Leave your
answers in simplest radical form.
B)
A)
C) Find the lengths of the legs of a 30-60-90
triangle with hypotenuse of length 12.
D) Find the lengths of the legs of a 30-60-90
triangle with hypotenuse of length 𝟒 𝟑.
Example 5
Using the Length of a Leg
Find the value of each variable. Leave your
answers in simplest radical form.
A)
B)
C) The shorter leg of a 30-60-90 triangle has
length 𝟔. What are the lengths of the other
two sides?
D) The longer side of a 30-60-90 triangle has
length 18. Find the lengths of the shorter leg
and the hypotenuse.
Homework:
7 – 3 Ditto; 1 – 13
Section 7 – 3
Continued…
Objectives:
To use the properties of 45-45-90 & 30-60-90
triangles
Example 6
Multi-Step Problems
A) The deer warning sign is an equilateral
triangle. Each side is 1 meter long. Find the area of
the sign.
B) A rhombus has 10-inch sides, two of
which meet to form a 30 degree angle. Find the
area of the rhombus.
C) A rhombus has 10-inch sides, two of
which meet to form a 60 degree angle. Find the
area of the rhombus.
Find the value of each variable. Leave your answer in simplest
radical form.
D)
E)
Find the area of each figure. Round to the nearest tenth.
F)
G)
Homework
Textbook Page 370 – 371; #25, 26, 27,
29, 34, 36, 38 , 39