Right Triangles

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