Mensuration

We are given an equilateral triangle PQR where the sides are divided into four, six, and eight equal parts respectively. The length of each subdivided part is an integer, so we need to find the minimum area of the triangle PQR.

Step 1: Determine the side lengths.
Let the side length of the equilateral triangle PQR be . The number of parts that each side is divided into is:
- is divided into 4 parts.
- is divided into 6 parts.
- is divided into 8 parts.
Since the length of each subdivided part must be an integer, the least common multiple (LCM) of 4, 6, and 8 will give the minimum integer side length that satisfies the conditions. The LCM of 4, 6, and 8 is 24. Therefore, the minimum side length of the triangle is .

Step 2: Calculate the area of the equilateral triangle.
The area of an equilateral triangle with side length is given by the formula: Substitute : Thus, the minimum area of the triangle PQR is cm², which corresponds to option (D).
Final Answer:

To calculate the distance from point Q to point O, we will break the problem down into steps based on the movements described:

1. From point O, you travel 3 units East to point P.
2. Then, from point P, you travel 4 units South.
3. Next, you travel in the North-East direction. The North-East direction implies equal displacement along the North and East axes. After travelling North-East, you end up 6 units East of point O.
4. Finally, you travel in the North-West direction. The North-West direction also involves equal displacement along the North and West axes, and you end up 8 units North of point P.

The distances form a right-angled triangle with the horizontal and vertical components, and by using the Pythagorean theorem, we find the straight-line distance between point Q and point O is 5 units.

Final Answer: 5