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
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.
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, 9-12-15, 30-40-50, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and there are many more examples
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
Legs are congruent
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: SohCahToa
Sine (sin): opposite leg / hypotenuse
Cosine (cos): adjacent leg / hypotenuse
Tangent (tan): opposite leg / adjacent leg
10*sin38 = x 10 * cos38 = y
y = 10 / cos42 10 * tan42 = x
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 500.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 120.His eye level is 6 feet above the top of the building.
How far is it from the car to his eyes?