Find the area of triangle ABC

Find the area of triangle ABC if AB = BC = 12 and angle ABC = 150 degrees.    

Since AB = BC triangle ABC is isoceles and angles BAC and BCA are equal 

Since ABC = 150^o and the sum of the interior angles of a triangle is 180^o  

then angles A and C total 30^o and therefore each of them is 15^o  

Draw a perpendicular from the apex B down to a point on side AC - call the point P  

                                                          BP  

Using the sine function:    sin(15) = ———  

                                                          12 

                                                BP = (12)(0.65029) = 7.80        this is the height of the triangle

                                                            AP

Using the cosine function:  cos(15) = ———  

                                                            12 

                                                 AP = (12)(0.96593) = 11.59       this is half the base of the triangle  

So the whole base of the triangle is (2)(11.59) = 23.18              (We could have skipped this step.  Do you know why?)  

                                                                1

The formula for the area of a triangle is  — • base • height  

                                                                 2  

So, the area of this triangle is 0.5 • 23.18 • 7.80 = 90.40 units²  

The answer in the back of the book might be slightly different, because of the rounding that happened here.  

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