We have been able to prove that the area of the triangle formed by the joining midpoints of triangle ABC is 1/4 the area of triangle ABC.
Area of triangle joining the points A, B & C is;
A₁ = ¹/₂[tex]\left[\begin{array}{ccc}1&1&1\\3&-2&5\\4&0&0\end{array}\right][/tex]
A₁ = ¹/₂(-1 * -20) + (1 * 8)
A₁ = 14
Thus;
Midpoint of AB is (¹/₂, 2)
Midpoint of BC is (³/₂, 0)
Midpoint of AC is (4, 2)
A₂ = [tex]\frac{1}{2} \left[\begin{array}{ccc}1&1&1\\\frac{1}{2} &\frac{3}{2} &4\\2&0&2\end{array}\right][/tex]
A₂ = ¹/₂(3 - 1(-7) + (1 * -3))
A₂ = ⁷/₂
Since A₁ = 14, we can equally say that;
A₂ = ¹/₄A₁
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