Let A B C and A B C^ be two non-congruent triangles with sides A B=4 , A C=A C^ =2 2 and angle B=30^ . The absolute value of the difference between the areas of these triangles is

Step-by-step solution

In A B C , by sine rule, aligned & a A = 2 2 30^ = 4 C & C=45^ , C^ =135^ & A=180^ - (45^ +30^ )=105^ aligned When C^ =135^ , then aligned & A=180^ - (135^ +30^ )=15^ & Area of A B C= 1 2 A B A C A & = 1 2 4 2 2 (105^ ) & =4 2 3 +1 2 2 =2( 3 +1) & Area of A B C^ = 1 2 A B A C A & = 1 2 4 2 2 (15^ )=2( 3 -1) aligned Difference of areas of triangles =|2( 3 +1)-2( 3 -1)|=4 A D=2, D C=2 Difference of areas of A B C and A B C^ aligned & = Area of A C C^ & = 1 2 A D C C^ = 1 2 2 4=4 aligned