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Clickers. Bellwork. A. B. Find the measure of angle DBC in square ABCD In the previous example, find the area of triangle BDC What is the length of segment BD What is the area of an equilateral triangle with side length 10

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bellwork

Clickers

Bellwork

A

B

Find the measure of angle DBC in square ABCD

In the previous example, find the area of triangle BDC

What is the length of segment BD

What is the area of an equilateral triangle with side length 10

What is the altitude(height) of an equilateral triangle with side length of 20

What is the altitude(height) of an equilateral triangle with side length of 32

x

C

D

bellwork solution

Bellwork Solution

A

B

Find m<DBC in square ABCD

In the previous example, find the area of triangle BDC

What is the length of segment BD

x

C

D

bellwork solution1

Bellwork Solution

What is the area of an equilateral triangle with side length 10

What is the altitude(height) of an equilateral triangle with side length of 20

What is the altitude(height) of an equilateral triangle with side length of 32

the concept
The Concept
  • Today we’re going to be working with some special right triangles that occur within other geometric figures
  • The ratios shown today will reappear throughout your odyssey throughout mathematics…
special triangle

Special Triangle

There are two special right triangles that occur naturally and are thus often studied in trigonometry. We’re going to look at one today and the other on Monday

What’s the length of the diagonal of a square of side length 1

1

This kind of isosceles triangle is called a 45-45-90 triangle because of the angles formed when you draw a diagonal across a square

1

special triangle1

Special Triangle

How would this relate to triangles whose sides are larger than 1?

5

Therefore the relationship between sides and hypotenuse this triangle is by radical two…

5

the theorem

The theorem

Theorem 7.8

In a 45-45-90 triangle, the hypotenuse is radical 2 times as long as each leg

on your own

On your own

Solve for x

on your own1

On your own

Solve for x

on your own2

On your own

Solve for x

on your own3

On your own

Solve for x

special triangle2

Special Triangle

Given an equilateral triangle of side length 2, can we determine the height of the triangle

2

h

This kind of triangle is called a 30-60-90 triangle because of the angles formed when you draw an altitude in an equilateral triangle

1

2

special triangle3

Special Triangle

How would this relate to triangles whose sides are larger than 1?

6

h

Therefore the relationship the 60o side and hypotenuse is one half and the other side is by radical three

3

6

the theorem1

The theorem

Theorem 7.9

In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

30o

60o

on your own4

On your own

Solve for x

on your own5

On your own

Solve for x

on your own6

On your own

Solve for x

on your own7

On your own

Solve for x & y

on your own8

On your own

Solve for x

on your own9

On your own

Solve for x

on your own10

On your own

Solve for x & y

homework

Homework

7.4 Worksheet

practical example

Practical example

The distance from Bill’s feet to his waist is 3 feet. While doing leg lifts at track practice, he wonders how high his feet are off the ground in order to get his mind off the burning pain in his abs.

How far off the ground are his feet when his legs make a 30o angle with the ground?

  • 1.5 ft
  • 2.60 ft
  • 5.20 ft
practical example1

Practical example

The distance from Bill’s feet to his waist is 3 feet. While doing leg lifts at track practice, he wonders how high his feet are off the ground in order to get his mind off the burning pain in his abs.

How far off the ground are his feet when his legs make a 45o angle with the ground?

  • 2.12 ft
  • 3 ft
  • 4.24 ft
practical example2

Practical example

The distance from Bill’s feet to his waist is 3 feet. While doing leg lifts at track practice, he wonders how high his feet are off the ground in order to get his mind off the burning pain in his abs.

How far off the ground are his feet when his legs make a 60o angle with the ground?

  • 1.5 ft
  • 2.60 ft
  • 5.20 ft
most important points

Most Important Points

  • 45-45-90 triangle side relationships
  • 30-60-90 triangle side relationships