Straight Lines
Triangle properties and midpoints
Grade 11
Question:
<p>If D, E and F are the middle points of BC, CA and AB respectively then the area of the triangle DEF is:</p><p>Consider 3 non-collinear points A(9, 3), B(7, –1) and C(1, –1).</p>
<p>(a) 12</p>
<p>(b) 6</p>
<p>(c) 4</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: The medial triangle (formed by midpoints of sides) has area exactly 1/4 of the original triangle.
<p><strong>Solution:</strong> The triangle formed by the midpoints of the sides of a triangle has area equal to 1/4 of the original triangle.</p><p>First, find the area of triangle ABC with vertices A(9, 3), B(7, –1), C(1, –1):</p><p>Area D = ½|9(–1 – (–1)) + 7(–1 – 3) + 1(3 – (–1))|</p><p>= ½|0 + 7(–4) + 1(4)|</p><p>= ½|–28 + 4| = ½|–24| = 12</p><p>Area of triangle DEF = D/4 = 12/4 = 3</p>
Correct Answer: D