The four triangles formed by joining the pairs of mid - points of the sides of a given triangle are congruent if the given triangle is :

Options

A. an isosceles triangle
B. an equilateral triangle
C. a right angled triangle
D. of any shape

Answer

To determine when the four triangles formed by joining the pairs of midpoints of the sides of a given triangle are congruent, we can follow these steps:

### Step 1: Understand the Problem
We start with a triangle ABC. We need to find out under what conditions the triangles formed by joining the midpoints of the sides of triangle ABC are congruent.

**Hint:** Think about the properties of triangles and how midpoints affect the sides.

### Step 2: Identify the Midpoints
Let D, E, and F be the midpoints of sides AB, BC, and CA respectively. By connecting these midpoints, we form four smaller triangles: ADF, BDE, CEF, and DEF.

**Hint:** Visualize or sketch the triangle and label the midpoints clearly.

### Step 3: Analyze the Congruence of Triangles
To check if triangles ADF, BDE, CEF, and DEF are congruent, we need to consider their sides and angles.

1. **Side Lengths:** Since D, E, and F are midpoints, the segments AD, DF, BE, and EF are all half the lengths of the corresponding sides of triangle ABC.
2. **Angles:** The angles at each vertex of the smaller triangles depend on the angles of triangle ABC.

**Hint:** Use the properties of congruent triangles (SSS, SAS, ASA) to analyze the relationships.

### Step 4: Determine Conditions for Congruence
For the triangles ADF and BDE to be congruent:
- They must have equal corresponding sides and angles.

This is guaranteed when triangle ABC is equilateral because:
- All sides are equal, and all angles are equal (60 degrees each).
- Thus, the triangles formed by the midpoints will also be equal in size and shape.

**Hint:** Consider the implications of having equal angles and sides in triangle ABC.

### Step 5: Conclusion
The four triangles formed by joining the pairs of midpoints of the sides of triangle ABC are congruent if triangle ABC is an **equilateral triangle**.

**Final Answer:** The given triangle must be an equilateral triangle.