Download 45-45-90 Special Triangle Discovery

Warm-Up:
Summary 1a + b bottom of page 8.
AND Simplify:
8
2
Homework Answers
72 + 152 < 212
45-45-90 Special Triangle
Discovery
We begin with a square with 1 unit on
each side. This is shown below.
Find the measurement of all the
angles and the lengths of each side.
Add a diagonal to the above square.
The diagonal creates two smaller
triangles in your above square.
Find the angle measures of the angles
in the two triangles.
Using the Pythagorean Theorem, find
the length of the diagonal
(or the hypotenuse of the right angle
triangle). KEEP IN RADICAL FORM.
We begin with a square with 2 unit on
each side. This is shown below.
Find the measurement of all the
angles and the lengths of each side.
Add a diagonal to the above square.
The diagonal creates two smaller
triangles in your above square.
Find the angle measures of the angles
in the two triangles.
Using the Pythagorean Theorem, find
the length of the diagonal
(or the hypotenuse of the right angle
triangle). KEEP IN RADICAL FORM.
We begin with a square with 3 unit on
each side. This is shown below.
Find the measurement of all the
angles and the lengths of each side.
Add a diagonal to the above square.
The diagonal creates two smaller
triangles in your above square.
Find the angle measures of the angles
in the two triangles.
Using the Pythagorean Theorem, find
the length of the diagonal
(or the hypotenuse of the right angle
triangle). KEEP IN RADICAL FORM.
Square with
___ Units
4
8
12
15
17
Side Length
of a Leg
Side Length
of a Leg
Side Length
of
Hypotenuse.
Is there a pattern you notice? Explain it.
Do you think this would work for all shapes (ie Rectangles,
triangles etc.)? Explain why or why not.
What effect does the size of the square have on the lengths of
the sides?
Conclusion: Given an “x” unit on each side square, what would
be the lengths of the sides, and the length of the hypotenuse?
Section 8.3 Notes: Special Right
Triangles
In a 45˚-45˚-90˚triangle, the legs 𝑙 are congruent
and the hypotenuse, ℎ is 2 times the
the length of a leg.
45˚-45˚90˚
Triangle *ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝑙𝑒𝑔 ∙ 2
Theorem *ℎ = 𝑙 2
Example 1: Find the measure of each
hypotenuse
a.
Example 1: Find the measure of each
hypotenuse
b.
Example 1: Find the measure of each
hypotenuse
You can also work backwards using theorem 8.8
to find the legs of a 45° – 45° – 90° triangle
a.
Example 1: Find the measure of each
hypotenuse
You can also work backwards using theorem 8.8
to find the legs of a 45° – 45° – 90° triangle
b.
Example 2.
The perimeter of a square is 48 meters. Find the
length of a diagonal.
Example 3.
The perimeter of a square is 20 cm. Find the
length of a diagonal.
You Try!
Find the value of x and y in each triangle.
You Try!
Find the value of x and y in each triangle.
You Try!
Find the perimeter of the given square.
You Try!