Chapter 8 – Right Triangles and Trigonometry
Section 1 – Geometric Mean
§ I can use the geometric mean to find parts of a right triangle with an altitude drawn to the hypotenuse.
§ I can use congruence criteria to solve problems about triangles and prove relationships in geometric figures.
· Geometric Mean – The geometric mean of 2 numbers “a” and “b” is the number “x” so that
1. Find the geometric mean between the numbers:
(a) 2 and 50 (b) 5 and 45 (c ) 12 and 15
***If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
2. Write the similarity statement identifying the three similar triangles in the figure.
(a) (b)
3. (a) Find c, d, and e. (b) Find x, y, and z.
(c) Find x, y, and z
4. (a) KITES: Mrs. Alspach is making a kite for her son. She has to arrange two support rods so that they are perpendicular. The shorter rod is 27 inches long. If she places the short rod 7.25 inches from one end of the long rod in order to form two right angles with the kite fabric, what is the length of the long rod?
(b) A community center needs to estimate the cost of installing a rock climbing wall by estimating the height of the wall. Sue holds a book up to her eyes so that the top and bottom of the wall are in line with the bottom edge and binding of the cover. If her eye level is 5 feet above the ground and she stands 11 feet from the wall, how high is the wall? Draw a diagram and explain your reasoning.
5. Find x, y, and z. 6. Find w
Homework – Page 535 – 538 (11, 15, 18-25, 27, 29, 31, 33, 35, 37) ODD & (49, 50)
Section 2 – Pythagorean Theorem and it’s Converse
· I can prove the Pythagorean Theorem using triangle similarity.
· I can use the converse of the Pythagorean Theorem to determine if a triangle is acute, obtuse, or right.
· I can list the common Pythagorean triples.
· I can use the Pythagorean Theorem to solve for unknown side length of a right triangle.
· I can apply geometric methods to solve design problems.
*Pythagorean Theorem can ONLY be used with a right triangle. The side across from the right angle is called the hypotenuse and labeled as “c”in the formula. The other 2 lines are called the legs and are labeled “a” and “b” (Page 542 shows some common Pythagorean triples you might want to know)
1. Find x
(a) (b)
2. Find x:
3. A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder?
Pythagorean Inequality Theorems:
(acute) à c2 < a2 + b2 and (obtuse) à c2 > a2 + b2
4, Determine whether each set of numbers can be the measures of the sides of a triangle. If so, classify the triangles as acute, right, or obtuse. Justify your answer
(a) 9, 12, and 15 (b) 10, 11, and 13 (c) 23, 42, 35
Homework – Page 546 – 548 (9 – 31)ODD (20, 39, 40, 41, 45, 49, 51)
Section 3 – Special Right Triangles
· I can solve right triangles including special right triangles (such as 30-60-90 and 45-45-90) by finding the measures of all sides and angles in the triangles.
**45-45-90 Triangle – these are used with squares lots of times. The formula uses the hypotenuse and legs
of the triangle. so
1. Find x:
(a) (b)
2. Find a:
**30-60-90 Triangle – Triangles include 3 types of lines. Hypotenuse, long side and short side.
3. Find x and y
(a) (b)
4. (a) QUILTING: A quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle?
(b) FURNITURE: The top of the aquarium coffee table shown is an isosceles right triangle. The table’s longest side, AC, measures 107 centimeters. What is the distance from vertex B to side AC? What are the lengths of the other two sides?
Homework – Page 556 – 559 (9 – 39)ODD (40, 46, 48)
Section 4 – Trigonometry
· I can use properties of similar right triangles to form the definitions of trigonometric ratios for acute angles.
· I can calculate sine and cosine ratios for acute angles in a right triangle when two side lengths are given.
· I can explain and use the relationship between sine of an acute angle and the cosine of its complement.
· I can draw right triangles that describe real world problems and label the sides and angles with their given measures.
SOH CAH TOA
· Sine – also known as sin of an angle and equals a side opposite an angle divided by the hypotenuse.
· Cosine – also known as cos of an angle and equals a side adjacent to an angle divided by the hypotenuse
· Tangent – also known as tan of an angle and equals a side opposite an angle divided by the side adjacent to the angle.
SIN = OPPOSITE/HYPOTENUSE COS = ADJACENT/HYPOTENUSE TAN = OPPOSITE/ADJACENT
SOH CAH TOA
1. Express each ratio as a fraction and a decimal to the nearest hundredth (2 decimal places).
(a) sin L (b) cos L (c) tan L
(d) sin N (e) cos N (f) tan N
2. Use a special right triangle to express the following functions as a simplifed fraction and decimal.
(a) sin 45 (b) sin 60 (c) sin 30
(d) cos 45 (e) cos 60 (f) cos 30
(g) tan 45 (h) tan 60 (i) tan 30
3. (a) The fitness trainer sets the incline on a treadmill to 7 degrees. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor?
(b) ARCHITECTURE: The front of the vacation cottage shown is an isoscles triangle. What is the height ‘x’ of the cottage above its foundation? What is the length ‘y’ of the roof? Explain your reasoning.
4. Use a calculator to find the measure of angle P to the nearest degree.
5. Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.
Homework – Page 567 – 570 (17, 19, 21, 29 – 45)ODD (51, 57, 59, 62) (Look over #’s 22-27)
Section 5 – Angles of Elevation and Depression
§ I can solve application problems involving right triangles, including angle of elevation and depression, navigation and surveying, using the Pythagorean Theorem and trigonometry.
examples of these pg.574
· Angle of Elevation - the angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line.
· Angle of Depression – the angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line.
1. At the circus, a person in the audience at the ground level watches the high-wire routine. A 5-foot-6 inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat’s head is 27 degrees?
2. Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52 degrees, what is the horizontal distance from the seal to the cliff, to the nearest foot?
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 degrees and 36 degrees. Find the distance between the two dolphins to the nearest meter.
**DO GUIDED PRACTICE EXAMPLES TOGETHER!
Homework – page 577-579 (4 – 15) and (20)
Section 6 – Law of Sines and Law of Cosines
§ I can derive the Law of Sines.
§ I can use the Law of Sines to solve real world problems.
§ I can derive the Law of Cosines.
§ I can use the Law of Cosines to solve real world problems.
§ I can distinguish between situations that require the Law of Sines or the Law of Cosines.
§ I can represent real world problems with diagrams of right and non-right triangles and use them to solve for unknown side lengths and angle measures.
Law of Sines:
Sin A = Sin B = Sin C
a b c
1. Find p. Round to the nearest tenth.
2. Find x. Round to the nearest degree.
Law of Cosines:
a2 = b2 + c2 -2bc cos A OR b2 = a2 + c2 – 2ac cos B OR c2 = a2 + b2 – 2ab cos C
3. Find x. Round to the nearest tenth.
4. Find the measure of angle L. Round to the nearest degree.
5. (a) From the diagram of the airplane shown, determine the approximate width of each wing. Round to the nearest tenth meter.
(b) LANDSCAPING: At 10 feet away from the base of a tree, the angle the top of a tree makes with the ground is 61°. If the tree grows at an angle of 78° with respect to the ground, how tall is the tree to the nearest foot?
6. Solve triangle PQR. Round the nearest degree. (be sure to solve for each angle!)
Homework – Page 587 – 589 (13 – 41)ODD (30, 47, 49)
Section 7 – Vectors
· Vector – a quantity that has both magnitude and direction.
· Magnitude – the length of the vector for its intial point to its terminal point. (Use distance formula to find the magnitude)
· Direction – the angle that is formed with the positive x-axis or any other horizontal line. (Use tangent of the angle where then endpoint is on the ray then add 180 to it)
· Standard Position – a vector that’s initial point is at the origin when placed on a coordinate plane.
· Component Form – a vector is described in terms of its horizontal change “x” and vertical change “y” from its initial point to its terminal point. The component form is listed as (x, y)
1. Write the component form of AB.
2. Find the magnitude and direction of ST for S (-3, -2) and T(4, -7)
3. Copy the vectors to find
(a) a – b (b) a + b (c) b – a
4. CANOEING: Suppose a person is canoeing due east across a river at 4 miles per hour. If the river is flowing south at 3 miles per hour, what are the resultant direction and speed of the canoe?
5. Add each set of vectors:
v = (-3, 5) x = (6, -4) y = (-3, -7) z = (1, 6)
(a) v + x
(b) y – z
(c) x + z
(d) v – y
Homework – Page 597 – 598 (8 – 19)ALL (21, 23, 25, 27) (28 – 33)ALL (35, 37)