Question

In a ΔABC, D, E and F are the mid-points of the sides AB, BC and CA respectively. Then the ratio of the area of a ΔDEF and the area of a ΔABC is:

• A. 1:4
• B. 1:2
• C. 2:3
• D. 4:5

Answer 1

The Correct Option is A

To solve this problem, we need to find the ratio of the area of triangle \(\Delta DEF\) to the area of triangle \(\Delta ABC\), where D, E, and F are the mid-points of the sides AB, BC, and CA of triangle \(\Delta ABC\) respectively.

Given:

1. D is the midpoint of AB
2. E is the midpoint of BC
3. F is the midpoint of CA

By the Midpoint Theorem in a triangle, it is known that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it. Therefore, the lines DE, EF, and FD will divide the triangle \(\Delta ABC\) into four smaller triangles of equal area.

Let's use the properties of medians and triangles: 

• The mid-segment of a triangle creates a smaller, similar triangle within the original triangle. Thus, \(\Delta DEF\) is similar to \(\Delta ABC\) with a ratio of similarity 1:2.
• The areas of similar triangles are in the square of their sides' ratio.

Thus, the ratio of the area of \(\Delta DEF\) to \(\Delta ABC\) is given by:

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

Therefore, the correct answer is 1:4, which matches Option:

1:4