Download Discovering 30-60-90 Special Triangles

Complete “You Try” section p.11 in your
workbook!
Homework Answers
Discovering 30-60-90 Special
Triangles
We begin with an equilateral triangle with 1 unit
on each side. This is shown below.
Find the measurement of all the angles
and the lengths of each side.
Construct an altitude from the top
vertex to the base in the above square.
The diagonal creates two smaller
triangles in your above triangle.
Find the angle measures of the angles in
the two triangles and the new
length of the base of the two triangles.
Using the Pythagorean Theorem, find
the length of the altitude
(or the hypotenuse of the right angle
triangle). KEEP IN RADICAL FORM.
We begin with an equilateral triangle with 2 unit
on each side. This is shown below.
Find the measurement of all the angles
and the lengths of each side.
Construct an altitude from the top
vertex to the base in the above square.
The diagonal creates two smaller
triangles in your above triangle.
Find the angle measures of the angles in
the two triangles and the new
length of the base of the two triangles.
Using the Pythagorean Theorem, find
the length of the altitude
(or the hypotenuse of the right angle
triangle). KEEP IN RADICAL FORM.
We begin with an equilateral triangle with 3 unit
on each side. This is shown below.
Find the measurement of all the angles
and the lengths of each side.
Construct an altitude from the top
vertex to the base in the above square.
The diagonal creates two smaller
triangles in your above triangle.
Find the angle measures of the angles in
the two triangles and the new
length of the base of the two triangles.
Using the Pythagorean Theorem, find
the length of the altitude
(or the hypotenuse of the right angle
triangle). KEEP IN RADICAL FORM.
Equilateral with
____ Units
4
8
10
12
16
Side length of
Hypotenuse
Side length of Base
Side Length of
Altitude
Do you see a pattern? Explain.
Conclusion: Given an equilateral triangle with
“x” unit on each side, what would be the lengths
of the sides, and the lengths of the altitude?
8.3: Special Right Triangles
Equilateral
Triangle
Relationship
A 30° – 60° – 90° triangle is another special right triangle.
You can use an equilateral triangle to find this relationship.
When the altitude is drawn from any vertex of an equilateral
triangle, two congruent 30° – 60° – 90° triangles are formed.
In the figure shown, ∆ 𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷, so 𝐴𝐷 ≅ 𝐶𝐷. If CD =
x, then AC = 2x. This leads to the next theorem.
30˚-60˚-90˚
Triangle
Theorem
In a 30˚-60˚-90˚ Triangle, the hypotenuse 𝒉 is 2
times the length of the shorter leg 𝒔 and the length
of the longer leg 𝒍 is 3 times the length of the
shorter leg.
*ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝑠ℎ𝑜𝑟𝑡 𝑙𝑒𝑔 ∙ 2
*𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔 = 𝑠ℎ𝑜𝑟𝑡 𝑙𝑒𝑔 ∙ 3
*ℎ = 𝑠 ∙ 2
*𝑙 = 𝑠 3
Example 4: Find x and y.
a)
Example 4: Find x and y.
b)
Example 4: Find x and y.
c)
Example 5:
A quilt has the design shown in the figure, in
which a square is divided into 8 isosceles right
triangles. If the length of one side of the square
is 3 inches, what are the dimensions of each
triangle?
Example 6:
Shaina designed 2 identical bookends according
to the diagram below. Use special triangles to
find the height of the bookends.
Example 7.
An equilateral triangle has a side length of 10
inches. Find the length of the triangles altitude.
Example 8.
The altitude of an equilateral triangle is 18
inches. Find the length of a side.
Summary!
Find all the missing side lengths. Leave
answers in simplified radical form.
Summary!
Find all the missing side lengths. Leave
answers in simplified radical form.
Homework Answers: WB p.30-31
1) 𝑥 = 8 2
2) x = 3
4) 𝑥 = 6 2
7) x = 1, y =
5) 𝑥 =
3
2
25 2
2
6) 𝑥 = 14 2
8) 𝑥 = 8 3, 𝑦 = 16
9) 𝑥 = 5.5, 𝑦 = 5.5 3
11) 𝑥 = 30 3
3) 𝑥 = 4 2
10) 𝑥 = 24 3
Exit Slip
Find the values of x and y.