Question

Hard! se as medidas dos angulos internos de um triangulo estao em progressao aritemtica e a medida do maior angulo e o quintuplo da medida do menor, entao a diferenca entre a medida do maior angulo e a soma das medidas dos outros dois e

Answer 1

$P.A.=(a_1,a_2,a_ 3);\qquad S_3=180;\qquad a_3=5a_1\\\\\\S_3=\dfrac{3(a_1+a_3)}{2}\\\\\\180=\dfrac{3(a_1+5a_1)}{2}\\\\\\180=\dfrac{3(6a_1)}{2}\\\\\\180=3(3a_1)\\\\180=9a_1\\\\a_1=\dfrac{180}{9}\\\\\\\boxed{a_1=20^\circ}$



$a_3=5a_1\\\\a_3=5(20)\\\\\boxed{a_3=100^\circ}\\\\\\\\S_3=a_1+a_2+a_3\\\\180=20+a_2+100\\\\180=120+a_2\\\\a_2=180-120\\\\\boxed{a_2=60^\circ}$



$a_3-(a_1+a_2)=\ ?\\\\100-(20+60)\Longrightarrow100-80=\boxed{\boxed{20^\circ}}$