Introduction:
Rangoli is a traditional Indian art form created during festivals and special occasions. It involves creating colorful designs on the floor using colored powders, rice, or flower petals. Shakhshi, a creative individual, prepared a Rangoli in the shape of a triangle during Diwali. The Rangoli consisted of a big triangle and a smaller triangle underneath it. Let's explore the area of the smaller triangle, PQR.

Explanation:

Step 1: Identify the triangles:
To understand the area of the smaller triangle, we need to identify the triangles involved in the Rangoli. In this case, we have two triangles:

1. Big Triangle: The larger triangle is the main shape of the Rangoli. Let's denote its vertices as A, B, and C.

2. Small Triangle: The smaller triangle is created underneath the big triangle. It is formed by connecting the midpoints of the sides of the big triangle. Let's denote its vertices as P, Q, and R.

Step 2: Properties of the small triangle:
Now, let's examine the properties of the smaller triangle, PQR:

1. Sides: The sides of PQR are formed by connecting the midpoints of the sides of the big triangle ABC. This means that PQ is the midpoint of AB, QR is the midpoint of BC, and PR is the midpoint of AC.

2. Length ratios: The sides of the small triangle are proportional to the sides of the big triangle. This means that PQ = 1/2 AB, QR = 1/2 BC, and PR = 1/2 AC.

Step 3: Area of the small triangle:
To find the area of the small triangle, we will use the formula for the area of a triangle, which is given by the formula:

Area = (1/2) * base * height

In the case of the small triangle PQR, the base and height can be determined as follows:

1. Base: The base of the small triangle is equal to PQ, which is 1/2 AB.

2. Height: The height of the small triangle is the perpendicular distance from the base to the opposite vertex, in this case, the height from R to PQ.

Step 4: Calculate the area:
Using the formula for the area of a triangle, we can calculate the area of the small triangle PQR as follows:

Area of PQR = (1/2) * PQ * height

Since PQ = 1/2 AB, and height is the perpendicular distance from R to PQ, we can substitute these values into the formula:

Area of PQR = (1/2) * (1/2 AB) * height

Conclusion:
In conclusion, the area of the smaller triangle PQR in Shakhshi's Rangoli can be determined by using the formula for the area of a triangle. By connecting the midpoints of the sides of the big triangle, we create the smaller triangle, and its area can be calculated using the lengths of the sides and the perpendicular height.