Special Right Triangles Unit 4

Simplifying Radicals √ radical Radicand – number inside the radical http://www.youtube.com/watch?v=HU5IawUD2o8 You can click on other videos for more explainations. √

Examples √6 ∙ √8 √2∙2∙2∙2∙3 4√3 2) √90 √2∙3∙3∙5 3√10 √6 ∙ √8 √2∙2∙2∙2∙3 4√3 2) √90 √2∙3∙3∙5 3√10 3) √243 √3 √3∙3∙3∙3∙3 9 √3 9

Division – multiply numerator and denominator by the radical in the denominator 4) √25 √3 5 ∙√3 √3 ∙√3 3 8 = √14 √ 28 7 6) √5 ∙ √35 √14 √5∙5 ∙7 √2∙7 5√7 √2∙7 √2∙7 √2∙7 35 √2 = 5 √2 14 2

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Pythagorean Theorem and It’s Converse Objective: to use the Pythagorean Theorem and it’s converse. c2 = a2 + b2

Key Concept Pythagorean Theorem c2 = a2 + b2 c - hypotenuse a – altitude leg b – base leg

Key Concepts Acute c2 < a2 + b2 Right c2 = a2 + b2 C b A Acute c2 < a2 + b2 Right c2 = a2 + b2 Obtuse c2 > a2 + b2 B a c C b A B a c C b A

Ex 1 Find the value of x. Leave in simplest radical form. Answer: 2 √11 x 12 10

Ex 2: Baseball A baseball diamond is a square with 90 ft sides. Home plate and second base are at opposite vertices of the square. About far is home plate from second base to the nearest foot? About 127 ft

Ex 3:Classify the triangle as acute, right or obtuse. 15, 20, 25 right b) 10, 15, 20 Obtuse

Pythagorean Triplet Whole numbers that satisfy c2 = a2 + b2. Example: 3, 4, 5 Can you find another set?

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Special Right Triangles Objective: To use the properties of 45⁰ – 45⁰ – 90⁰ and 30⁰ – 60⁰ - 90⁰ triangles.

Isosceles Right Triangle Key Concept 45⁰-45⁰ -90⁰ X – X – X2 102 22 2 10 2 10

Example 1 Find x. Simplify. 45 – 45 – 90 x – x - x 2 8 – 8 – 82 82

Try 1: FIND THE MISSING LENGTHS. SIMPLEST RADICAL FORM. 45⁰ 45⁰ 90⁰ X X X2 1212 X2 = 1212 2 2 X = 121 MISSING SIDE LENGTHS ARE 121. 1212 45⁰

Example 2 Find x. Simplify. 45 – 45 – 90 y – y - y 2 – – 28 28 = y 2 set up equation 2 2 divide by root 2 y = 28 2 2 = 28 2 2 = 142

Try2: Find the length of the hypotenuse of a 45⁰ – 45⁰ – 90⁰ triangle with legs of length 5√6 . 45⁰ – 45⁰ – 90⁰ x - x - x√2 X = 5√6 x√2 = 5√6√2 substitute into the formula = 10 √3

Try 3 Find the length of a leg of a 45⁰ – 45⁰ – 90⁰ triangle with hypotenuse of length 22. 45⁰ – 45⁰ – 90⁰ x - x - x√2 x√2 = 22 solve for x X = 22 = 22√2 = 11√2 √2 2

Try 4: The distance from one corner to the opposite corner of a square field is 96ft. To the nearest foot, how long is each side of the field? 45⁰ – 45⁰ – 90⁰ x - x - x√2 x√2 = 96 solve for x X = 96 = 96√2 = 48√2 √2 2

What is the relationship of the legs and hypotenuse of an isosceles right triangle? 45⁰ – 45⁰ – 90⁰ x - x - x√2

Equilateral Triangle Key Concept 30⁰ - 60⁰ - 90⁰ X – X3 – 2X 30⁰ 4 2 3 60⁰ 2

Example 3 Solve for missing parts of each triangle: x = 10 y = 5√3 x y

Example 4 The longer leg of a 30⁰ – 60⁰ - 90⁰ triangle has length of 18. Find the lengths of the shorter leg and the hypotenuse. 30⁰ – 60⁰ - 90⁰ x - x√3 - 2x x√3 = 18 solve for x X = 18 = 18√3 = 6√3 – short leg √3 3 12√3 - hypotenuse

Try 5 30 - 60 - 90 X - X3 - 2X 9 2X = 9 X = 9/2 a = 9/2 b = 93 2 30 - 60 - 90 X - X3 - 2X 25 25 = x 3 3 3 X = 25 3 3 d = 25 3 c = 50 3 3

What is the relationship of the 30-60-90 right triangle? 30⁰ – 60⁰ - 90⁰ x - x√3 - 2x

Exit Ticket X = 5 Y = 5 2 Z = 53 3 W = 103

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Similarities in Right Triangles Objective: To find and use relationships in similar right triangles

Geometric mean with similar right triangles

Type 1 Relationship: altitude side2 side 1 altitude = altitude Side 1

Example 1

Try 1:

Type 2 relationship: Hypotenuse Leg(hyp) Leg ______ side = Big Small

Example 2:

Example 3:

Try: 2

Try 3:

Try 4:

Exit Ticket Find x, y, and z. X = 6 9 x 36 = 9x 4 = x 9 = z z 9+x y = x 9+x y Y ² = 4(13) Y = 2√13

8-3 and 8-4 Right Triangle Trigonometry Objective To use sine, cosine and tangent ratios to determine side and angle measures in triangles

Key Concept Sine (sin) Cosine(cos) Tangent(tan) SOH CAH TOA The Old Aunt Sat On Her Coat and Hat

Example 1 Find sin E, cos E, and tan E. sin E = 8/10 = 4/5 Try: Find sin F, cos F, and tan F. sin F = 6/10 = 3/5 cos F = 8/10 = 4/5 tan F =6/8 = 3/4 E 6 10 G 8 F

Example 2 – round to the nearest tenths. OPP/ADJ Tan (37⁰) = x/3 3(tan (37⁰)) = x X 2.3 Opp adj

Try 1 OPP/HYP Sin (67⁰) = 10/x 10/sin (67⁰) = x X  10.9 Opp hyp

TRY 2 Adj/hyp Cos (40⁰) = 6/x 6/cos(40⁰) = x X 4.6 adj hyp

Ex 3:When solving for angles use inverse. Round to the nearest degree Adj/hyp Cos(x) = 8/18 Cos-1(8/18)  64⁰

TRY Opp/adj Tan (x) = 11/8 tan-1(11/8)  54⁰

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Angles of Elevation and Depression Objective: To use angles of elevation and depression to solve problems

Try Round to the nearest tenths. Opp/adj Tan(25⁰) = x/250 250(tan(25⁰)) = x x 116.6 ft

Try round to the nearest hundredth Opp/hyp Sin (40⁰) = 30/x 30/sin(40⁰) = x X 46.67 ft

Try Opp/adj = tan α Tan(35⁰) = x / 30 30(tan(35⁰)) = x X 52 5 + 52 = 57ft X 5ft 35⁰ 30 ft

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Law of Sines Objective: To derive the Law of Sines To apply the law of sines to solve problems

Law of Sines Sin A Sin B Sin C a b c When to use: Oblique triangles given: AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side = =

Example 1: Find the missing angle and sides. The angles in a ∆ total 180°, so angle C = 30°. Set up the Law of Sines to find side b: A C B 70° 80° a = 12 c b

Continued A C B 70° 80° a = 12 c b = 12.6 30°

Try 1 Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm 30° A C B 115°

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Law of Cosines Objective: To derive the Law of Cosines To apply the law of cosines to solve problems

A (bcos C, bsinC) b c (0, 0) B a C (a, 0)

A b c A (bcos C, bsinC) B a C b c (0, 0) B a C (a, 0) (bcos C, bsinC)

(a, 0) (bcos C, bsinC) C A B c b a

Law of Cosines

Law of Cosines When you use it: SAS – when you know two sides and an included angle and you want to find the third side SSS – when you know three sides and you want to find an angle

Example 1 A Find the missing sides and angles 7 c 30 C B 4

Try 1: A Find the missing sides and angles. c 21 123º B C 18

C The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet in length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)? A

Example 2: Find the missing angles. C B a = 30 c = 15 b = 20

Try 2:

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Objective: To use and apply vectors to solve applied problems.

How to draw and read a vector.  V  U Tail Head O  OA A

How to Draw the Sum of Vectors   U + V  V  U O  U  OA A

Try: Draw the sum of and  U  V  V  U

How to sketch a vector on a coordinate plane

Try: Sketch the vectors below.

How to find the magnitude of a vector (Simplest radical form)

How to find the direction of a vector To find the angle we will need to use Trigonometric ratios. tan (α) = 15/10 Tan-1(15/10)  56⁰ 15√13 15 10

Try: Resultant (magnitude and direction)

How do you find the direction and magnitude of a vector How do you find the direction and magnitude of a vector? What is still confusing?