Geometry Pythagorean Theorem of Right Triangles Angles of Elevation and epression Law of Sines and Law of osines Pythagorean Theorem Recall that a right triangle is a triangle with a right angle. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. hypotenuse The sides form that right angle are the legs. The side opposite the right angle is the hypotenuse. The hypotenuse is also the longest side.
Example: Solve for x: Find the length of the missing side of the right triangle. 81 + 144 = x -15 is a extraneous solution, a distance Is the missing side a leg or the hypotenuse of the right triangle? Example: Find the length of the missing side of the right triangle. Is the missing side a leg or the hypotenuse of the right triangle? Find the length of the missing side.
Find the length of the missing side. The safe distance of the base of the ladder from a wall it leans ladder should be 7 feet from the wall. How far up the wall will a ladder reach? Solve using a 2 2 2 The dimensions of a high school basketball court are 84' long and 50' wide. What is the length from one corner of the court to the opposite corner? A NBA court is 50 feet wide and the length from one corner of the court to the opposite corner is (Round the answer to the nearest whole number) 94 feet The Pythagorean Theorem can also be used in figures that contain right
Find the perimeter of the square. Start here: 2 2 2 note: Before finding the perimeter of the of each side. Find the area of the triangle. By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. the height of the triangle. Find the perimeter of the rectangle. 120 square feet 46 square inches 46 square feet
Find the perimeter of the square. (Round to the nearest tenth) onverse of the Pythagorean Theorem If the square of the longest side of a triangle is equal to the right triangle. Theorem If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. Remember longest side
Theorem lassify the triangle as acute, right, or obtuse. If the square of the longest side of a triangle is less than sum of the squares of the other two sides, then the triangle not a triangle not a triangle not a triangle
Review acute triangle right triangle, then triangle is right., then triangle is obtuse., then triangle is acute. How many do you know? Pythagorean Theorem. How? B click click
The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. Given: is a right angle B Given Given Reasons B Prove: is a right angle ef of Perp Lines. 2 lines that form a rt angle All rt angles are Reflexive Prop of is a right angle AA~ ef of Perp Lines All rt angles are Reflexive Prop of 30 30 30 30 AB ~ B AA~ 60 30 60 60 60 60 Reasons d c b e a c b a b d a e Given: Prove: c d AB is a right triangle. is an altitude. b e a (1) Altitude of a rt triangle theorem. efinition of similar triangles. Using the multiplication multiply the equation by bc. simplify Altitude of a rt triangle theorem. efinition of similar triangles. Using the multiplication multiply the equation by ac (2) simplify Reasons Using the addition property Given: Prove: AB is a right triangle. is an altitude. istributive Property Given Substitution Find the length of the altitude KI? c e a Simplify d b
x x x 12 x x 15 Which ratio is the ratio of corresponding sides? 16 of two positive numbers a What is an arithmetic mean? The sum of n values divided by the number of values (n). given a rectangle with sides a and b, find the side of the square whose area equals that of the rectangle. For more information click on this link: Arithmetic Mean vs Geometric Mean
Find the geometric mean of 8 and 14. 17 Find the geometric mean of 7 and 56. Write the answer is simplest radical form. A B (only the positive value) 18 Find the geometric mean of 3 and 48. divides the the hypotenuse into two segments. The B
19 Find x. 100 20 Find x. that is adjacent to the leg B B B
21 Is PR a geometric mean between QR and SR? 22 23 24 Find y. 5 16 25 Find y. 26 Find x. 3 18 24 None of the above
two special right triangles. triangle is an isosceles right triangle, where the hypotenuse is 2 times the length of the leg. Find the length of the missing sides. Write the answer in simplest radical form. hypotenuse = leg( an you prove this? Find the length of the missing sides of the right triangle. Find the length of the missing sides.
27 Find the value of x. 5 28 Find the value of y. 5 29 30 This can be proved using an equilateral triangle. right triangle, the hypotenuse is twice the length of the shorter leg and the longer leg is 3 times the length of the shorter leg. let a = x, c = 2x and b= B c=2x B b hypotenuse = 2(shorter leg) longer leg = 3(shorter leg) A a=x
Find the length of the missing sides of the right triangle. opposite the smallest angle and the longest side is opposite the largest angle. HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF Find the length of the missing sides of the right triangle. Example The altitude (or height) divides the triangle into two 30 The length of the shorter leg is 7 ft. The length of the longer leg is 7 60 84.87 square ft
31 Find the value of x. 2)/2 32 Find the value of x. 33 Find the value of x. 34 35
hypotenuse = 2(2.5) hypotenuse = 5 feet and rises at angle of 30 36 A skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45. What is the height of the ramp? Round to the nearest hundredth. 37 What is the length of the base of the ramp? Round to the nearest hundredth. 38 39
Ever since the construction of the tilted south and is at risk of falling below 83 degrees, it is feared the tower will collapse. Engineers can measure the angle of slant using Engineers very carefully measure the perpendicular distance from a tower window Then they measure the distance from the tower to points, E or G. F A WHY? B E Let's calculate the ratio's of the height to the base for each right triangle. Triangle Height Base Ratio Height / Base AB A=50m B=5m 50/5=10 BE E=30m BE=3m 30/3=10 FBG FG=20m BG=2m 20/2=10 lick for interactive website to investigate. Notice that all of the ratios are the same. WHY? The ratio of height/base is also called the slope ratio (rise/run) or tangent ratio.
is the reference angle, the side opposite is the side adjacent is b. and the hypotenuse is c. hyp opp hyp opp When is the reference angle, the side opposite is b. the side adjacent (or next to) B is a. and the hypotenuse is c. 40 What is the side opposite to J? 41 What is the hypotenuse of the triangle? 42 What is the side adjacent to J? 43 What is the side opposite K? A JL B LK KJ
44 What is the side adjacent to K? A JL B LK KJ two sides of a Trigonometric Ratios is the ratio of the sine sin The 3 Trigonometric Ratios adjacent side This spells... SOHAHTOA or θ which is a pneumonic to help you remember the sides of a right triangle (you'll need to remember the spelling). Example Example opp adj hyp 6 8 4 6 3 adj opp hyp 8 6 3 8 4
45 What is the sin R? 46 What is the cosr? A 9/13 B 7/9 7/13 47 What is the tanr? 48 What is the sinq? A 9/13 B 7/9 7/13 9/7 49 What is the cosq? A 9/13 B 7/9 7/13 9/7 50 What is the tanq? A 9/13 B 7/9 7/13 9/7
The angle of slant of the Tower of Pisa is 84.3 To find the trigonometric ratio of an angle, use a calculator or a trig table. heck that your calculator is set for degrees (not radians) and round your answer to the ten thousandth place (4 decimal places). Find the following: sin 84.3 =.9951 click cos 84.3 =.0993 click tan 84.3 = 10.0187 click A F B angle of slant 51 Evaluate sin 60. Round to the nearest ten thousandth. A 0.5 B 0.8660 1.7321 0.5774 52 Evaluate cos 60. Round to the nearest ten thousandth. A 0.5 B 0.8660 1.7321 0.5774 53 Evaluate tan 60. Round to the nearest ten thousandth. A 0.5 B 0.8660 1.7321 0.5774 Trig tables were used by early mathematicians and astronomers to calculate distances that they were unable to measure directly. Today, calculators are usually used. x opp adj
A A 54 Using B, which is the correct trig equation needed to solve for x. A 55 solve for x. 56 solve for x.
57 Using K, which is the correct trig equation needed to solve for x. Using your calculator, find the tan 84.3 Round your answer to 4 decimal places. opp You can rewrite 10.0187 with a denominator of 1 and use the cross product property or multiply both sides of the equation by 5 using the multiplication property of equality (see next slide). adj opp Multiply both sides of the equation by 5 using the multiplication property of equality. adj Round your answer to the nearest hundredth. Round your answer to the nearest hundredth.
58 59 Find the length of LP. Round your answer to the nearest tenth. Explain and use the relationship between the sine and cosine of complementary angles. The sum of the interior angles of any triangle is equal to 180 degrees. A and B are complementary angles. omplementary angles are two angles whose sum of their measures is 90 degrees. The acute angles of a right triangle are always complementary. 60 For right triangle AB, what is the measure of B? A 30 degrees B 50 degrees 60 degrees cannot be determined
61 If the, find the complementary angle? A 20 degrees B 70 degrees 160 degrees none of the above In a right triangle, the acute angles are complementary. m A + m B = 90 53.1 + 36.9 = 90 sin A = 4/5 sin 53.1 =.7997 cos B = 4/5 cos 36.9 =.7997 sin A = cos B sin 53.1 = cos 36.9 The sine of an angle is equal to the cosine of its complement. cos A = 3/5 cos 53.1 =.6004 sin B = 3/5 sin 36.9 =.6004 cos A = sin B cos 53.1 = sin 36.9 The cosine of an angle is equal to the sine of its complement. sine function cosine function 62 Given that sin 10 =.1736, write the cosine of a complementary angle. A sin 10 =.1736 B sin 80 =.9848 Sine and osine are called co-functions of each other. o-functions of complementary angles are equal. cos 10 =.9848 cos 80 =.1736 63 Given that cos 50 =.6428, write the sine of a complementary angle. A sin 50 =.7660 B sin 40 =.6428 cos 50 =.6428 cos 40 =.7660 64 Given that cos 65 =.4226, write the sine of a complementary angle. A sin 25 =.4226 B cos 25 =.9063 sin 65 =.9063 cos 65 =.4226
65 What can you conclude about the sine and cosine of 45 degrees? solve a right triangle means to find all 6 values in a triangle. The and the x y z x y 26 y z z
Try this... 26 13.48 11 z x x z z y y You will need to use the If sinθ =, θ = sin -1 If cosθ =, θ = cos -1 Pronounced inverse sine, inverse cosine, and inverse tangent. The 3 Inverse Trigonometric Ratios θ = ( ) θ = θ = adjacent side If tanθ =, θ = tan -1 θ With the sine, cosine and tangent trig functions, if you know the angle θ and the measure of one leg, then you can find the measure of a leg of a triangle. With the inverse sine, inverse cosine and inverse tangent trig functions, if you know the measures of 2 legs of a triangle, you can find the measure of the angle. θ Remember:
66 67 the angle measure 68 the angle measure θ adj hyp θ θ θ θ A A
θ 69 Which is the correct trig equation to solve for θ A A B 70 Which is the correct trig equation to solve for 71 Which is the correct trig equation to solve for A B A B Try this... 72 Find E.
73 Find m. 74 75 76 77 Find the m P. 78 Find RT.
trigonometric ratios to solve word problems involving angles of elevation and depression? 79 How can you describe the angle relationship between the angle of elevation and the angle of depression? A corresponding angles B alternate interior angles alternate exterior angles none of the above
Amy is flying a kite at an angle of 58 sin x How high is the kite off the ground? x = 134 3 feet Now, we must add in Amy's arm height. 134 + 3 = 137 You are standing on a mountain that is 5306 feet high. You look do.5774x = 5312 x Try this... elevation is 55 80 If you are 5.5 feet tall, how far are you from the base of the tree?
81 82 Law of Sines and Law of osines The Law of Sines and Law of osines can be used to solve any triangle. You can use the Law of Sines when you are given - 1. Two angle measures and any side length (AAS or ASA) 2. Two side lengths and the measure of a non-included angle (SSA) when the angle is a right angle or an obtuse angle. The Law of Sines has a problem dealing with SSA when the angle is acute. There can be zero, one or two solutions. lick on: for more info. You can use the Law of osines when you are given - 3. Three side lengths (SSS) 4. Two side lengths and the measure of an included angle (SAS) Law of Sines then sin A sin a b c Given: AB with side lengths a, b, and c raw an altitude from to side AB Prove: sin A sin Let h be the length of the altitude a b c h Reasons click Given ef of Altitude click click ef of sine sin A sin a b c Multiply click by b. Mult Prop of =. To use the Law of Sines, Multiply click by a. Mult Prop of =. Substitution click Prop of = ivide by ab. click ivision Prop of =
h Given: Prove: sin A sin a b c raw an altitude from B to side A Let g be the length of the altitude g Reasons click ef of Altitude ef clickof sine sin A sin a b c Select the ratios based on the given information. Given: m B, m and BA (side c) (AAS) Which ratios must be used first? Why? Multiply click by c. Mult Prop of =. Multiply click by a. Mult Prop of =. Substitution click Prop of = ivide by ac. ivision Prop of = click Substitution click Prop of = First we can find the length side b. find the Triangle Sum Theorem m A + m B + m = 180 o Now we find the length side a. Try this... sina a sin c hint
Example... 83 Find the m A. 84 Which ratio must be used to find the length of b or 85 What is the length of b? 86 What is the length of c? Law of osines If AB has sides of length a, b, and c, then: To use the Law osines, you must be given the (SSS) angle (SAS). sides
h Given: Prove: (similar reasoning shows that ) AB with side lengths a, b, and c Given click raw an altitude from to side AB. Let h be the length of the alt. Let x be the length of A. Then (c-x) is the length of B. In A, cosa = x/b (1) x=b(cosa) (2) In A, In B, Reasons click ef of Altitude Segment click Addition Postulate efinition click of cosine Multiply click by b. Mult Prop of =. Pythagorean click Theorem Pythagorean click Theorem click Simplify click Substitution, equation (2) Associative click Prop of Addition Substitution, click equation (1) The formula you choose depends on which angle you To find the m A, 36.22 = 23 + 27 256 = 529 + 729-1242(cosA) 256 = 1258-1242(cosA) -1002 = -1242(cosA).8068 = cosa A = cos -1 (.8068) m A 36.22 = 16 + 27 529 = 256 + 729-864(cosB) 579 = 985-864(cosB).4699 = cosb B=cos -1 (.4699) or Using 2 different methods, the answers are slightly different because of rounding. Try this... 61.97 36.22 To find the m, Use the Triangle Sum Theorem.
87 88 8 9 8 9 89 Which formula would you use to find the m<a? 90 91 What is the m? 92 What is the measure of B (ASA)? B A
93 The Law of Sines and osines is used to solve... acute triangles all triangles Find the area of the triangle. b is the base of the triangle b = 10. h is the altitude (or height). It is the perpendicular bisector of the base in an isosceles triangle. Find the area of the triangle. 67.38 67.38
Given: Prove: 94 Which of the following expressions can be used to find the area of the triangle below? Select all that apply. Reasons h AB with side lengths a, b, and c Let h be the length of the altitude Given A ef of sine B Multiply by a. Mult Prop of =. efinition. Formula for the area of a triangle. F Substitution Prop of = ommutative Prop of Multiplication 95 Find the area of the triangle to the nearest tenth. 96 Find the area of the triangle to the nearest tenth. 97 Find the area of the triangle to the nearest tenth.