Question

If A (-1,5), B (3,1) , c (5,7), are the vertices of triangle ABC , and D,E,F are the mid points of BC , CA and AB. Prove that the area of triangle ABC is four times the area of triangle DEF.

Answer 1

A(-1,5) B(3,1) C(5,7) 1. (-1,5) (3,1) Use formula x₁ +x₂/2. And y₁ +y₂/2 1,3 2.(3,1)(5,7) 4,4 3.(5,7)(-1,5) 2,6 F(1,3) D(4,4) E(2,6) Δ=1/2∡|x₁(y₂-y₃)+x₂(y₃-y₂)+x₃(y₁-y₂)| Δ=1/2|4(6-3)+2(3-4)+1(4-6)| Δ=1/2|4(3)+2(-1)+1(-2)| ∆=1/2|12-2-2| ∆=1/2|10-2| ∆=1/2|8| ∆=1/2*8 ∆=4 A(-1,5)B(3,1)C(5,7) ∆=1/2|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂) ∆=1/2|-1(-1+7)+3(7-5)+5(5-1)| ∆=1/2|6+6+20| ∆=1/2|12+20| ∆=1/2|32| ∆=1/2*32 ∆=16