chapter 7 right triangles and trigonometry

__Section 7.2 The Pythagorean Theorem and Its Converse__

**Pythagorean Theorem: **In
a right triangle, the sum of the squares of the legs equals the square of the
hypotenuse.

The legs are represented as a and b, the hypotenuse is represented as c.

There are two shortcuts that can usually be used that involve the Pythagorean Theorem:

**To find the Hypotenuse (c):**

1. square the legs

2. Add them together

3. Take the square root of the sum

**To find a Leg (a or b):**

1. square the hypotenuse and the known leg

2. Subtract them

3. Take the square root of the difference

**Example:**

**Example:**

**Converse of the** **Pythagorean
Theorem: **If the sum of the squares of two sides equals the square of
the third side, then the triangle is right.

So if , then the triangle is right.

**Example:**

Is this a right triangle?

**Pythagorean Triples**:
three whole numbers that represent the sides of a
right triangle.

Some common examples of Pythagorean triples:

3-4-5, ** 6-8-10**,

**Can these three lengths be the sides of a
right triangle?**

**Then state whether they form a Pythagorean
triple.**

**10, 15, 20
9, 40, 41
3/8, 4/8, 5/8**

__Area of Right Triangles__

Area = Base (*one leg*) * height (*other leg*) / 2

__Section 7.3 Special Right Triangles__

**45-45-90 Triangle**:

**
OR**

Legs are congruent

**30****-60-90
Triangle**:

**Examples:**

**Other Examples:**

The perimeter of an equilateral triangle is 60 inches. Find the length of an altitude.

The diagonal of a square is 14. Find the length of a side and then find the perimeter of the square.

The altitude of an equilateral triangle is 10. Find the length of a side and then find the perimeter of the equilateral triangle.

__Section 7.4 Trigonometry__

**Trig Ratios: ****S****oh****C****ah****T****oa**

**Sine **(sin): * o*pposite
leg /

**Cosine **(cos): ** a**djacent leg /

**Tangent **(tan): ** o**pposite leg /

**Example:**

**10*sin38 =
x
10 * cos38 = y**

**Example:**

**y = 10 /
cos42
10 * tan42 = x**

**Example:**

**y = 8 /
sin64
x = 8 / tan64**

__Section 7.4 (part 2)__

Trig can be used to find the angle measures of a right triangle as well.

Use the inverse of the sine or cosine or tangent to calculate the measure of an angle.

__ Solve the Right Triangle__
(find all unknown measures)

__Section 7.5 Angles of Elevation and Depression__

**Angle of Elevation: **Angle
between line of sight and horizontal when looking upward.

**Angle of Depression: **Angle
between line of sight and horizontal when looking downward.

**Example 1:** The angle
of elevation from point A to the top of a cliff is 34 degrees. If point A
is 1000 feet from the base of the cliff, how high is the cliff?

**Example 2: **The
angle of depression from the top of an 80 foot building to point B on the ground
is 42 degrees. How far from the base of the building is point B?

**Example 3: **The
tailgate of a truck is 3.5 feet above the ground. A loading ramp is
attached to the tailgate with an incline of 10 degrees. Find the length of
the ramp.

**Example 4: **A
sledding hill is 300 yards long with a vertical drop of 27.6 yards. Find
the angle of depression of the hill.

**Percent Grade is calculated by dividing the
rise (vertical elevation) by the run (horizontal distance)**

**Example 5: **
What is the angle of elevation of these hills?

Steepest street in the world 35% in New Zealand

Green slopes are usually between 6% and 25% grade

Blue slopes are between 25% and 40%

Black slopes are greater than 40%

What is the angle of depression range for each slope?

Ian is standing on the ground 60 feet from a cliff looking up to the top
with an angle of elevation of 50^{0. }^{His eye
level is 6 feet above ground.}

How tall is the cliff?

Now Ian is standing on top of a 40 foot building looking down at Teren with an
angle of depression of 12^{0. }^{His eye level
is 6 feet above the top of the building. }

How far is it from the car to^{
his eyes}?