In an equilateral triangle ABC , D is the mid-point of AB and E is the mid-point of AC. Find the ratio between ar (` triangleABC ) : ar(triangleADE)`

Answer

To find the ratio between the area of triangle ABC and the area of triangle ADE, we can follow these steps:

### Step 1: Understanding the Triangle
Given an equilateral triangle ABC, we know that all sides are equal and all angles are 60 degrees. D is the midpoint of side AB, and E is the midpoint of side AC.

### Step 2: Applying the Midpoint Theorem
By the midpoint theorem, the line segment DE (connecting the midpoints D and E) is parallel to the side BC of triangle ABC. This means that triangles ADE and ABC are similar.

### Step 3: Establishing Similarity
Since DE is parallel to BC, we can say that:
- Angle DAE = Angle BAC (common angle)
- Angle ADE = Angle ABC (corresponding angles)
- Angle AD = Angle ACB (corresponding angles)

Thus, by the Angle-Angle-Angle (AAA) criterion for similarity, triangle ABC is similar to triangle ADE.

### Step 4: Ratio of Corresponding Sides
Since the triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Let’s denote:
- The length of side AB = c (since ABC is equilateral, all sides are equal)
- The length of side AD = c/2 (since D is the midpoint of AB)

### Step 5: Finding the Ratio of Areas
The ratio of the sides is:
\[ \frac{AD}{AB} = \frac{\frac{c}{2}}{c} = \frac{1}{2} \]

Now, the ratio of the areas of the triangles is given by the square of the ratio of their corresponding sides:
\[ \frac{ar(\triangle ADE)}{ar(\triangle ABC)} = \left(\frac{AD}{AB}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 6: Finding the Final Ratio
To find the ratio of the area of triangle ABC to the area of triangle ADE, we take the reciprocal:
\[ \frac{ar(\triangle ABC)}{ar(\triangle ADE)} = \frac{1}{\frac{1}{4}} = 4 \]

Thus, the ratio of the area of triangle ABC to the area of triangle ADE is:
\[ \text{Ratio} = 4 : 1 \]

### Final Answer
The ratio between the area of triangle ABC and the area of triangle ADE is \( 4 : 1 \).

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