In Chapter 4, you investigated similarity and discovered that similar triangles have special relationships. In this chapter, you will discover that the side ratios in a right triangle can serve as a powerful mathematical tool that allows you to find missing side lengths and missing angle measures for any right triangle. You will also learn how these ratios (called trigonometric ratios) can be used in solving problems.

In this chapter, you will learn: when side lengths don’t form a triangle which sides are longer in a triangle and why common Pythagorean triples special triangles how the tangent ratio is connected to the slope of a line the trigonometric ratios to find missing measurements in right triangles how to find angles in a right triangle given side lengths how to find the area of regular polygons

5.1 Is The Answer Reasonable? Pg. 2 Triangle Inequality

5.1 – Is The Answer Reasonable? Triangle Inequality You now have several tools for describing triangles (lengths, areas, and angle measures), but can any three line segments create a triangle? Or are there restrictions on the side lengths of a triangle? And how can you know that the length you found for the side of a triangle is accurate? Today you will investigate the relationship between the sides of a triangle.

5.1 – CONSTRUCTING TRIANGLES Consider the segments below. Construct a triangle with the given side lengths.

3.5 in

5.2 – CONSTRUCTING TRIANGLES Consider the segments below. Construct a triangle with the given side lengths.

3 in

5.3 – CONSTRUCTING TRIANGLES Consider the segments below. Construct a triangle with the given side lengths.

2.5 in

5.4 – IS IT POSSIBLE? a. Use the manipulative provided by your teacher to investigate what is happening in the previous problem. Can a triangle be made with any three side lengths? If not, what condition(s) would make it impossible to build a triangle? Try building triangles with the side lengths provided by your teacher.

Trial #1: ______________ Trial #2: ______________ Trial #3: ______________ Make an equilateral triangle yes Make an isosceles triangle yes Make a scalene triangle yes

Trial #4: ______________ Trial #5: ______________ Trial #6: ______________ Make a triangle with sides of green, yellow, and blue (8.66cm, 10cm, 12.24cm) yes Make a triangle with sides of orange, purple, and red (5cm, 7.07cm, 14.14cm) no Make a triangle with sides of 2 orange and a yellow (5cm, 5cm, 10cm) no

b. For those triangles that could not be built, what happened? Why were they impossible? The sum of two sides needs to be greater than the third a b c a + b > c a + c > b b + c > a

c. Determine if the following lengths could be made into a triangle. Support your answer. Trial #7: 3cm, 6cm, 10cm Trial #8: 4cm, 9cm, 12cm Trial #9: 2cm, 4cm, 5cm Trial #10: 3cm, 5cm, 8cm 9 > 10 no 13 > 12 yes 6 > 5 yes 8 > 8 no

ll.html?movie=/hotmath_help/gizmos/trian gleInequality.swf

5.5 – MAXIMUM AND MINIMUM LENGTHS Examine the pictures of the triangles below. There is a range of values that will complete a triangle. The fact that there are restrictions on the side of a triangle is referred to as the Triangle Inequality Theorem. Determine the minimum and maximum values that will make a triangle. What value does it have to be above? What value does it have to be below?

x + 13 > 19 x + 19 > > x x > 6 x > > x x < 32 More than 6 Less than 32 6 < x < 32

15 – 14 <x< <x< 29

16 – 13 <x< <x< 29

21 – 9 <x< <x< 30

b – a <x< b + a

5.6 – TRIANGLE IMPOSSIBLE Is it possible to construct a triangle with the given side lengths? If not, explain why not. a.3, 4, 5b.1, 4, 6 c. 17, 17, 33 d. 7, 52, 45 7 > 5 yes 5 > 6 no 34 > 33 yes 52 > 52 no

5.7 – SMALLEST SIDE Use the information to determine what is the smallest whole number the following can be:

5.7 – SMALLEST SIDE Use the information to determine what is the smallest whole number the following can be:

5.8 – PERIMETER A student draws a triangle with a perimeter of 12in. The student says that the longest side measures 7in. How do you know that the student is incorrect? 7in P = 12in+5in