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Right Triangles

Right Triangles Introduction to the Mathematics for Solving Unknown Angles and Leg Lengths of Right Triangles. Lesson Topics. Triangle Classifications Right Triangles Angle Sum Theorem Triangle Congruency Postulates Right Triangle Similarity Theorem Pythagorean Theorem

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Right Triangles

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  1. Right Triangles Introduction to the Mathematics for Solving Unknown Angles and Leg Lengths of Right Triangles

  2. Lesson Topics • Triangle Classifications • Right Triangles • Angle Sum Theorem • Triangle Congruency Postulates • Right Triangle Similarity Theorem • Pythagorean Theorem • Opposite and Adjacent Sides • Sine, Cosine, & Tangent Ratios

  3. Triangles A triangle is polygon having three sides and three angles. Triangles are classified according to two characteristics: • Congruency Between Side Lengths • Angle Sizes

  4. Triangle Classifications Classifying Triangles According to Angle Sizes An angle (pronounced “theta”) isdefined as the space between two intersecting segments. Angles are measured in degrees. • Right Triangle: A triangle that has a 90° angle. • Obtuse Triangle: A triangle with one angle that is greater than 90°. • Acute Triangle: A triangle in which all the angles are less than 90°.

  5. Triangle Classifications Classifying Triangles According to Congruency Between Side Lengths Note: Congruent means equal in size. • Scalene Triangle: A triangle with no congruent sides. • Isosceles Triangle: A triangle with at least two congruent sides. • Equilateral Triangle: A triangle with three congruent sides.

  6. Right Triangles A right triangle contains a 90° angle (also called a right angle). Depending on the lengths of its sides, a right triangle may be classified as either scalene or isosceles. Note: A “square box” that appears on the inside corner of a triangle signifies a right angle.

  7. Right Triangles The sides of a triangle are sometimes referred to as “legs.” The hypotenuse of a right triangle is the side that occurs opposite the right angle, and is always the longest leg. b a  

  8. Angle Sum Theorem TheAngle Sum Theorem for Triangles states the sum of the three angles of any triangle is 180°. Because two of the three angles of the right triangle below are known, the value of angle X can be calculated. 1 + 2 + 3 = 180° X + 40° + 90° = 180° X + 130° = 180° X = 50°

  9. Triangle Congruency Postulates Two triangles are congruent (z) if and only if there is a correspondence between their vertices such that… • each pair of corresponding sides is congruent • each pair of corresponding angles is congruent AND

  10. Triangle Congruency Postulates The following congruency postulates can be used to determine if two triangles are congruent to each other… • Angle-Angle-Side (AAS) • Angle-Side-Angle (ASA) • Side-Side-Side (SSS) • Side-Angle-Side (SAS)

  11. Right Triangle Similarity Theorem Two triangles are considered similar (~) if… • their vertices can be matched so that corresponding angles are congruent (); or… A A′, B B′, andC C′

  12. AB BC AC .5 or .5 : 1 = = = A′B′ B′C′ A′C′ 1 Right Triangle Similarity Theorem Two triangles are considered similar (~) if… • the ratios of the lengths of corresponding sides are equal

  13. X 3 10 4  4X = 3  10 = 30 4 X = Right Triangle Similarity Theorem Problem: Given the two similar triangles at left, calculate the value of X. X = 7.5

  14. Pythagorean Theorem Pythagorean Theorem: the squared length of a right triangle’s hypotenuse is equal to the sum of the squared lengths of its legs. a2 + b2 = c2 32 + 42 = 52

  15. Pythagorean Theorem Pythagorean Theorem: the squared length of a right triangle’s hypotenuse is equal to the sum of the squared lengths of its legs. a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25 25 = 25

  16. Pythagorean Theorem The Pythagorean Theorem can be used to calculate the length of side X on the right triangle below. a2 + b2 = c2 32 + X2 = 52 9+ X2 = 25 X2 = 16 X = √16 X = 4

  17. Opposite and Adjacent Sides The terms “opposite” and “adjacent” are used to identify specific legs of a right triangle relative to the location of one of the acute angles.

  18. Length of the Opposite Leg Length of the Hypotenuse sin  = opposite hypotenuse sin  = .60 1 3 5 Sine Ratio The Sine (abbrev. sin) of an angle in a right triangle is a ratio between the length of the side opposite that angle, and the length of the hypotenuse. sin 36.87° = = = .60

  19. opposite hypotenuse 29 .53 sin  = sin 32° =  .53 = X = 29 X 29 X Sine Ratio Problem: Calculate the length of side X. X = 54.72

  20. Length of the Adjacent Leg Length of the Hypotenuse cos  = adjacent hypotenuse cos  = .80 1 cos 36.87° = = 4 5 = .80 Cosine Ratio The Cosine (abbrev. cos) of an angle in a right triangle is a ratio between the length of the side adjacent to that angle, and the length of the hypotenuse.

  21. adjacent hypotenuse cos  = cos  = = .59375   = cos-1 .59 19 32 Cosine Ratio Problem: Calculate the value of .  = 53.84°

  22. Length of the Opposite Leg Length of the Adjacent Leg tan  = .75 1 3 4 Tangent Ratio The Tangent (abbrev. tan) of an angle in a right triangle is a ratio between the lengths of the sides that are opposite and adjacent to that angle. opposite adjacent tan  = tan 36.87° = = = .75

  23. opposite adjacent tan  = tan 25° =  X = .47 x 30 .47 = X 30 X 30 Tangent Ratio Problem: Calculate the length of side X. X = 14.10

  24. opposite hypotenuse sine  = 3 5 Sine, Cosine, & Tangent Ratios If the lengths of the legs of a right triangle change, but the angles remain constant, the Sine, Cosine, and Tangent ratios remain constant. Example: = .60 sine 36.87° = 6 10 sine 36.87° = = .60

  25. Solving Right Triangles Given two values of a right triangle, the other four values can now be found. This is called solving a right triangle. Example 1: 42° Find the missing values on the right triangle pictured at right. 8 in

  26. Solving Right Triangles First, label the unknowns. Remember that the hypotenuse is always labeled as c. while unknown angles are generally labeled with Greek letters. • In this example,  will be the easiest value to find, since we know that the angle-sum of any triangle is always 180. • + 42 = 90 • = 90 – 42 • = 48° 42° c 8 in  a

  27. Length of the Opposite Leg Length of the Hypotenuse sin  = c = 8 0.743 Solving Right Triangles Now, we use one of the trigonometric ratios discussed earlier. 42° Sin 48° =  0.743 = c 8 in = 10.767 in 48° a

  28. Solving Right Triangles Since we know two of the legs, we can use the Pythagorean theorem to find the length of the third side. a2 + b2 = c2 42° a2 + 82 = 10.7672 10.767 in 8 in a2 + 64 = 115.928 a2 = 51.928 48° a = = 7.21 in a

  29. Solving Right Triangles Example 2: Find the missing values on the right triangle pictured at right. 5 in 9 in

  30. 106 = 10.3 = c Solving Right Triangles Label the unknowns. Call the hypotenuse c and the missing angles  and .Because we have two sides, we can find the hypotenuse using the Pythagorean Theorem. a2 + b2 = c2  52 + 92 = c2 c 5 in 25 + 81 = c2  106 = c2 9 in

  31. Length of the Opposite Leg Length of the Adjacent Leg tan  = Solving Right Triangles Using one of the trigonometric functions, set up an equation with one of the unknown angles.  Then, tan  = 10.3 in 5 in tan  = 1.8 tan-1  = 1.8  = 60.95° 9 in

  32. Solving Right Triangles Because the sum of the acute angles = 90°, we have: 90 – 60.95 =  29.05° =  We have found all missing values. The right triangle has been solved. 60.95° 10.3 in 5 in  9 in

  33. References Billstein, R., Libeskind, S., & Lott, J. W. (2004). A problem solving approach to mathematics (8th ed.). Boston, MA: Pearson Education, Inc. Clemens, S. R., O’Daffer, P. G., Cooney, T. J., & Dossey, J. A. (1990). Addison-Wesley geometry. Menlo Park, CA: Addison-Wesley. Serra, M. (1997). Discovering geometry: An inductive approach (2nd ed.). Berkeley, CA: Key Curriculum Press. Soanes, C. & Hawker, S. (Eds.). (2005). Compact Oxford English dictionary (5th ed.). NY: Oxford University Press.

  34. Credits: Writer: Wanda T. Staggers Lesson Editor: Ed Hughes Narration: CJ Amarosa PLTW Editor: Production: CJ Amarosa

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