Question

In the\( ∆ABC; D, E\) and \(F\) are the mid points of the sides \(BC, CA\) and \(AB.\) Then area of \(∆DEF:\) area of \(∆ABC=\)_____

• A. 1:4
• B. 4:1
• C. 1:3
• D. 3:4

Answer 1

The Correct Option is A

To solve the problem, we need to determine the ratio of the area of \( \triangle DEF \) to the area of \( \triangle ABC \), where \( D \), \( E \), and \( F \) are the midpoints of the sides \( BC \), \( CA \), and \( AB \), respectively.

1. Understanding the Midpoints and Medians:
When \( D \), \( E \), and \( F \) are the midpoints of the sides of \( \triangle ABC \), the segments \( AD \), \( BE \), and \( CF \) are medians of the triangle. The medians of a triangle intersect at the centroid, which divides each median into a ratio of 2:1.

2. Properties of the Median Triangle:
The triangle formed by connecting the midpoints of the sides of a triangle (i.e., \( \triangle DEF \)) is called the **median triangle** or **midpoint triangle**. A key property of this triangle is that its area is exactly one-fourth of the area of the original triangle (\( \triangle ABC \)). This can be proven using similarity and area ratios.

3. Similarity of Triangles:
Since \( D \), \( E \), and \( F \) are midpoints, the sides of \( \triangle DEF \) are parallel to and half the length of the corresponding sides of \( \triangle ABC \). Therefore, \( \triangle DEF \) is similar to \( \triangle ABC \) with a similarity ratio of \( \frac{1}{2} \).

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding side lengths. Since the similarity ratio is \( \frac{1}{2} \), the area ratio is:

\[ \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]

This means the area of \( \triangle DEF \) is \( \frac{1}{4} \) of the area of \( \triangle ABC \).

4. Final Answer:
The ratio of the area of \( \triangle DEF \) to the area of \( \triangle ABC \) is \( 1:4 \).

Final Answer:
The ratio is \( {1:4} \).