Geometry Notes - Chapter 7
Right Triangles and Trigonometry
Warm-Up
1. Solve each proportion. 2. Determine the product.
a. b.a. b. c.
7.1 Geometric Mean
The Geometric Mean between two numbers is
1. Find the geometric mean between:
a. 2 and 50b. c.
2. The geometric mean between 8 and x is 4. Find x
3. The geometric mean between x and 5 is 10. Find x
Right Triangles and their altitude drawn from the vertex of the right angle
2 In ABC, AD = 4 and CD = 12. Find BD. /3. Find x,y, and z.
/ 4. Find the measure of RT, QT and ST if
QS = 17, and RS = 5.
5. In QRS, RT = 8, QT = 5, find TS, RQ, QS
6. To find the height of her school building, Anna held a book near her eye so that the top and bottom of the building were in line with the edges of the cover. If Anna's eye level is 5 feet above the ground and she is standing about 10.25 feet from the building, how tall is the building? Round to the nearest tenth. /
7.1 Review
Find x, y, and z
7.2 The Pythagorean Theorem and Its Converse
Warm-Up
1. Find the measure of the hypotenuse of each right triangle having legs with the given measure. Round to the nearest hundredth, if necessary.
a. 5 and 12b. 6 and 8c. 15 and 15d. 14 and 27
Pythagorean Theorem:
Converse of Pythagorean Theorem:
2. Find x.
.
3. Carly and her father are building an addition on their home.
A section of a new wall is 6 feet wide and 8 feet high. They are
installing a diagonal brace to make sure that the vertical boards
are perpendicular to the horizontal boards. How long does the
brace need to be?
4. Verify that a triangle with verticesX(-4, -3), Y(1, 1), Z(9, -9) is a right
triangle. /
Pythagorean Triple:
5. Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple.
a.10, 15, 18
b.24, 45, 51
c.
d. , 8, 8
7.3 Special Right Triangles
Warm-Up
Simplify each expression in exact form (no decimals)
1. 2. 3. 4. 5. 6.
Properties of 45-45-90 Triangles
Examples:
1. Find x
a./ b.
/ c.
e.
/ f.
/ g.
2. A junior baseball league uses a baseball diamond in which the bases
are placed 60 feet apart. Find the distance from 2nd base to home plate.
3. Triangle TJK is a 45-45-90 triangle with right angle J. Locate the coordinates of point T in Quadrant II for J(-2, -3) and K(3, -3). /4. Find the perimeter of a square 5. Find the diagonal of a square
with diagonal 12 centimeters. with perimeter 28 meters.
Properties of 30-60-90 Triangles
3. Find x and y
a./ b.
/ c.
e.
/ f.
/ g.
7.3 Review:
1./ 2.
/ 3. ABC is equilateral with
side length = 10
4.
/ 5
/ 6.
7.4 Trigonometry
The study of trigonometry comes from two Greek terms, trigon, meaning ______and metron, meaning ______
A ratio for the lengths of sides of a right triangle is called a ______
The three most common trigonometric ratios are ______, ______and ______
Examples:
1. Find sin A, cos A, tan A, sin B, cos B, and tan B.Express each ratio as a fraction and as a
decimal. Round to the nearest hundredth. /
2. Use a calculator to find each value to the nearest ten thousandth.
a.cos 48°b. sin 85°c. tan 37°
e. cos(x) = f. sin(x) = g. tan(x) =
3. Find x. Round to the nearest tenth.
4. A plane in flight begins to climb at a constant angle of 2° for the next 70 ground miles. Find the change in the plane's altitude to the nearest tenth mile.
5. Find mQ in right trianglePQR for P(4, 4), Q(13, 1), and R(3, 1). /
7.5 Angle of Elevation and Depression
Vocabulary:
The angle between the line of sight and the
horizontal when an observer looks upward
is called the ______.
The angle between the line of sight and the
horizontal when an observer looks downward
is called the ______.
Examples:
1. The angle of elevation from point A to the top of a hill is 49. If point A is 400 feet from the base of the hill, how high is the hill? /2. Find the angle of elevation of the sun when a 12.5 meter tall telephone pole casts an 18 meter long shadow.
3. The angle of depression from the tip of a sheer cliff to point A on the ground is 35. If point A is 280 feet from the base of the cliff, how tall is the cliff?
4. The angle of depression from a balloon on a 75-foot string to a person on the ground is 36. How high is the balloon?
1. A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower. That person observes an air traffic controller at the window of a 132 foot tower. What is the angle of elevation?
2. From the top of a 120 foot high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19. How far from the base of the tower is the airplane?
3. Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon’s position is 154 meters above sea level, and the angles of depression to the two dolphins are 35° and 36°. Find the distance between the two dolphins to the nearest meter.
4. A pilot is flying 10,000 feet and wants to take the plane up to 20,000 feet over the next 50 miles. What should be his angle of elevation to the nearest tenth? (Hint: There are 5280 feet in a mile.)
5. Find the angle of elevation of the sun when a 7.6 meter flagpole casts an 18.2 meter shadow. Round to the nearest tenth of a degree.
7.6 - The Law of Sines
Vocabulary:
In trigonometry, when a triangle is ______the ______
and the ______can be used to find missing parts of triangles.
Examples:
1. Find b. Round to the nearest tenth
2. Find p. Round to the nearest tenth
3. Solve ABC if mA = 33, mB = 47, and b = 14. Round angle measures to the nearest degree and side measures to the nearest tenth.
4. Solve DEF if mD = 112, mF = 8, and f = 2. Round angle measures to the nearest degree and side measures to the nearest tenth.
5. When the angle of elevation to the sun is 62, a telephone pole tilted at an angle of 7 from the vertical casts a shadow of 30 feet long on the ground. Find the length of the telephone pole to the nearest tenth of a foot.
7.7 - The Law of cosines
Given the lengths of the sides of a triangle, or two sides and an angle that is not opposite a given side, use the ______to solve the triangle.Examples:
1. Find a if c = 8, b = 10, and mA = 60.
2. Find x if y = 11, z = 25, and mX = 45.
3. Find mQ, mR, mS /4. Solve LMN if LM = 5, MN = 24, LN = 27. Find mL
5. Solve DEF /Page 1