The triangles inside `triangleABC` , shown above, are formed by joining the midpoints of the sides and then repeating the process. If a point is chosen at random inside `triangleABC` ,what is the probability that the point lies in the shaded region ?

Answer

To solve the problem of finding the probability that a randomly chosen point inside triangle ABC lies in the shaded region (the smallest triangle formed by repeatedly joining the midpoints of the sides), we can follow these steps:

### Step 1: Understand the process of forming triangles
We start with triangle ABC. By joining the midpoints of the sides of triangle ABC, we form a new triangle, which we will denote as triangle DEF. This process can be repeated indefinitely, creating smaller triangles within the previous triangles.

**Hint:** Visualize the process of forming triangles by joining midpoints to better understand the relationships between the areas.

### Step 2: Calculate the area of triangle DEF
The area of triangle DEF is one-fourth of the area of triangle ABC. This is because when we join the midpoints of a triangle, the new triangle formed has sides that are half the length of the original triangle's sides. Therefore, the area scales by the square of the ratio of the sides.

**Hint:** Remember that area scales with the square of the linear dimensions. If the sides are halved, the area is reduced by a factor of \( (1/2)^2 = 1/4 \).

### Step 3: Calculate the area of triangle HIG
Continuing this process, the area of triangle HIG (formed by joining the midpoints of triangle DEF) will also be one-fourth of the area of triangle DEF. Thus, the area of triangle HIG will be:
\[ \text{Area of HIG} = \frac{1}{4} \times \text{Area of DEF} = \frac{1}{4} \times \frac{1}{4} \times \text{Area of ABC} = \frac{1}{16} \times \text{Area of ABC} \]

**Hint:** Keep track of how each new triangle's area relates to the previous triangle's area.

### Step 4: Calculate the area of triangle PQR
Similarly, the area of triangle PQR (formed by joining the midpoints of triangle HIG) will be:
\[ \text{Area of PQR} = \frac{1}{4} \times \text{Area of HIG} = \frac{1}{4} \times \frac{1}{16} \times \text{Area of ABC} = \frac{1}{64} \times \text{Area of ABC} \]

**Hint:** You can see a pattern emerging: each new triangle's area is one-fourth of the area of the triangle before it.

### Step 5: Determine the probability
The probability that a randomly chosen point inside triangle ABC lies within the shaded region (triangle PQR) is the ratio of the area of triangle PQR to the area of triangle ABC:
\[ \text{Probability} = \frac{\text{Area of PQR}}{\text{Area of ABC}} = \frac{1/64 \times \text{Area of ABC}}{\text{Area of ABC}} = \frac{1}{64} \]

**Hint:** The probability is simply the ratio of the area of the shaded region to the area of the entire triangle.

### Final Answer
Thus, the probability that a randomly chosen point inside triangle ABC lies in the shaded region is:
\[ \boxed{\frac{1}{64}} \]