Day 101 – Area of a triangle

Introduction We have been knowing that the area of a triangle is given by 1 2 ×𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡 where height here implies the perpendicular height. That is true, however, there are situations where this may be of little help. That arises when the perpendicular height is not known. We would like to use the concept of trigonometry to see if we can have a solution to such situation. In this lesson, we are going to derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. We will then solve problems using the formula.

Vocabulary Acute triangle A triangle where all interior angles are acute angles Obtuse triangle A triangle where one of the interior angles is an obtuse angle.

Area of a triangle when the included angle is acute Consider following triangles ABC, an acute and obtuse triangles. The area of the triangle is Area = 1 2 ×𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡 = 1 2 × 𝐶𝐴 × 𝐷𝐵 = 1 2 𝑏ℎ We now express ℎ in terms of 𝜃 and 𝑎 since ℎ is not always given. a c b B C A h 𝜃 D a c b B C A h 𝜃 D

Using trigonometric ratios, sin 𝜃 = ℎ 𝑎 Thus, ℎ=𝑎 sin 𝜃 Substituting ℎ, we get = 1 2 𝑏ℎ= 1 2 𝑎𝑏 sin 𝜃 Area of a triangle when the included angle is obtuse a c b B D A h 𝜃 C

Angle 𝐷𝐶𝐵=180−𝜃 Thus, sin 180−𝜃 = ℎ 𝑎 Thus, ℎ=𝑎 sin (180−𝜃) But, sin 180−𝜃 = sin 𝜃 if 𝜃 is an acute angle. The proof of this is beyond the scope of this lesson. Hence ℎ=𝑎 sin (180−𝜃) =𝑎 sin 𝜃 Area of the triangle is = 1 2 × 𝐶𝐴 × 𝐷𝐵 = 1 2 𝑏ℎ= 1 2 𝑎𝑏 sin 𝜃 Thus, the area of a triangle, in both cases is 1 2 𝑎𝑏 sin 𝜃 = 1 2 𝑎𝑏 sin 𝐶 Where 𝑎 and 𝑏 are sides of the triangle and 𝐶 or 𝜃 the included angle.

Example Find the area of a triangle of sides 5 𝑖𝑛, 8 𝑖𝑛 and included angle of 73°. The area is given by Area= 1 2 𝑎𝑏 sin 𝐶 where 𝑎 and 𝑏 are sides of the triangle and 𝐶 the included angle. Area= 1 2 ×5×8× sin 73 =19.13 𝑠𝑞. 𝑖𝑛

homework In what situation, with reference to the angle, could we have an area given be 1 2 𝑎𝑏 sin 𝜃 = 1 2 𝑎𝑏.

Answers to homework The included angle, 𝜃, is a right triangle.

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