A life not lived for others is not a life worth living. Albert Einstein
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1 life not lived for others is not a life worth living. lbert Einstein
2 Sides adjacent to the right angle are legs Side opposite (across) from the right angle is the hypotenuse. Hypotenuse Leg cute ngles Leg
3 In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. b c a 2 + b 2 = c 2 a
4 Find the length of the hypotenuse. 16 x 30
5 Find the value of x. 1) 2) x x
6 The top of a ladder rests against a wall, 23 feet above the ground. The base of the ladder is 6 feet away from the wall. What is the length of the ladder?
7 Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.
8 Pythagorean triple is a set of three positive integers that satisfy a 2 + b 2 = c 2. 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17 x2 6, 8, 10 10, 24, 26 14, 48, 50 16, 30, 34 x3 9, 12, 15 15, 36, 39 21, 72, 75 24, 45, 51 x4 12, 16, 20 20, 48, 52 28, 96, , 60, 68 x5 15, 20, 25 25, 60, 65 35, 120, , 75, 85
9 Find the unknown length x 8 x x
10 ad is never good until worse happens. Danish Proverb
11 If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. b c If c 2 = a 2 + b 2, then is a right triangle. a
12 Tell whether the given side lengths of a triangle can represent a right triangle. 1) 5, 6, 7 2) 10, 11, 12 3) 5, 10, 15 4) 16, 12, 20 5) 9, 11, 15
13 Determine if the triangle is a right triangle D D D 22.5 D
14 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is a acute triangle. b c If c 2 < a 2 + b 2, then is a acute triangle. a
15 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is a obtuse triangle. b c If c 2 > a 2 + b 2, then is a obtuse triangle. a
16 Determine if the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse. 1) 5, 6, 7 2) 10, 11, 12 3) 5, 10, 15 4) 16, 12, 20 5) 9, 11, 15
17 To determine if a wall is perpendicular to the ground you measure the height of a wall to be 8 feet and 7 feet around the ground. What would the length of a string from the ceiling to the base have to be in order for the wall to be perpendicular to the ground?
18 Things could be worse. Suppose your errors were counted and published every day, like those of a baseball player. non.
19 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. D D D
20 Identify the three similar right triangles. D
21 Find the value of the variable. Round decimal answers to the nearest tenth x D 10
22 Refer to chapter 6 section 1. D D = D D D = D D D = D
23 Fill in the remainder of the proportions. 1) D =? 2)? = D D 3) = D D? D
24 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. D D = D D D
25 Find the value of the variable x D
26 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. D = = D D
27 Find the value of the variables. 2 x z 8 D y
28 The superior man(/woman) blames himself(/herself), the inferior man(/woman) blames others. Don Shula
29 In a triangle, the hypotenuse is 2 times as long as each leg. Hypotenuse = leg 2 x x 2 x
30 Find the lengths of the missing sides in the triangles
31 Fill in all known information for the right triangle
32 In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Hypotenuse = 2 shorter leg Longer leg = shorter leg 3 x 2x x 3
33 Find the lengths of the missing sides of the triangles
34 You are trying to make a logo for a group you are involved in. The shape you made is roughly a equilateral triangle. You know that each edge of the logo will be 8 in. and you need to calculate how much material will be needed for the logo. What is the area of the logo you have designed? (How much material will you need?)
35 I think we consider too much the good luck of the early bird, and not enough the bad luck of the early worm. Franklin D. Roosevelt
36 trigonometric ratio is a ratio of the lengths of two sides in a right triangle
37 The tangent trigonometric ratio is a ratio of the lengths of legs of a right triangle. The tangent is specifically the ratio opposite leg to adjacent leg tan = or tan = tan = or tan = 20 15
38 s seen below, the tangent ratios of complementary angles are reciprocals of each other. We can also use the same right triangle because the acute angles of a right triangle are complementary tan = or tan = tan = or tan = 20 15
39 Setup the tan and the tan. (another way this question is asked is to find the tan and tan. )
40 Find the value of the side lengths of the triangles. D F 7 30 E
41 What is the tan 45 and the tan 60?
42 Using a sextant you measure the angle of elevation to the top of a flagpole to be 40. You also know you are standing 35ft from the flagpole. How tall is the flagpole?
43 Doing what s right isn t the problem. It s knowing what s right. Lyndon. Johnson
44 The sine trigonometric ratio is a ratio of the lengths of the opposite leg and hypotenuse of a right triangle. The sine is specifically the ratio of opposite leg to hypotenuse sin = or sin = sin = or sin = 20 25
45 Find the sin and the sin. (setup the sin ratio for angle and angle )
46 Find the value of the variables. D G 12 I y 47 F x E H
47 The cosine trigonometric ratio is a ratio of the lengths of the adjacent leg and hypotenuse of a right triangle. The cosine is specifically the ratio of adjacent leg to hypotenuse cos = or cos = cos = or cos = 15 25
48 Find the cos and the cos. (setup the cos ratio for angle and angle )
49 Find the value of the variables. D G I 25 y F x 35 E H
50 n angle of elevation is an angle measured from the horizontal and rises from the horizontal. n angle of depression is an angle measured from the horizontal and sinks from the horizontal. ngle of depression ngle of Elevation
51 The angle of elevation for a ramp is 5. The ramp is 18 ft. long. What is the height of the ramp?
52 Find all information about this triangle. J 6ft K 15 L
53
54 To solve a right triangle means to find the measure of all of its sides and angles. To be able to solve a right triangle you need either: Two side lengths of a right triangle. One side length and one acute angle measure of a right triangle.
55 Solve the right triangle. J K 6ft 15 L
56 J 6ft K 12ft L
57 The inverse trigonometric ratios produce the angle that matches up to ratio given by the trigonometric ratios. tan = or tan = sin = or sin = cos = or cos = What is the measure of angle?
58 tan = or tan = m = tan 1 or m = tan
59 sin = or sin = m = sin 1 or m = sin
60 cos = or cos = m = cos 1 or m = cos
61 Solve the right triangle. J 6ft K 12ft L
62 Find the measure of angle
63 Suppose your company is building a ramp for entry into your building. n angle of elevation less than 10 is required by state law (not sure if this is the actual requirement). You have 25ft available outside your office to build the ramp and the doorway is 5ft from ground level. Would a ramp built here satisfy the state requirement?
64
65 For any triangle the following statement of proportionality is true. (Note: in this case c does NOT have to be the biggest side.) sin a sin = b = sin c c a b
66 Find the value of the variables b 80 a
67 For any triangle the following equation is true. (Note: in this case c does NOT have to be the biggest side.) c 2 = a 2 + b 2 2ab cos c b a
68 Find the value of the variables. 17 x y a
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