Let's go through the problems one by one:

41. The perimeter of a square is 70 cm. Its area (cm²) is:

• The perimeter $P$ of a square is given by $P = 4s$, where $s$ is the side length. $s = \frac{70}{4} = 17.5 \, \text{cm}$
• The area $A$ of a square is $A = s^2$: $A = (17.5)^2 = 306.25 \, \text{cm}^2$
• Answer: B. 306.25

42. What is the apothem of a regular polygon having an area of 225 sq.cm. and a perimeter of 60 cm?

• The area $A$ of a regular polygon is given by $A = \frac{1}{2} P a$, where $P$ is the perimeter and $a$ is the apothem. $225 = \frac{1}{2} \times 60 \times a$ Solving for $a$: $225 = 30a \quad \Rightarrow \quad a = \frac{225}{30} = 7.5 \, \text{cm}$
• Answer: D. 7.5

43. Find the area of the inscribed circle of a regular hexagon whose side has length 8.

• The radius of the inscribed circle is the same as the apothem of the regular hexagon, which is given by $r = \frac{\sqrt{3}}{2} s$, where $s = 8$: $r = \frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3}$
• The area of the circle is $A = \pi r^2$: $A = \pi \times (4\sqrt{3})^2 = \pi \times 48 = 48\pi$
• Answer: B. 48π

44. The median of a triangle is a line drawn from a vertex:

• A median connects a vertex of the triangle to the midpoint of the opposite side.
• Answer: D. To the midpoint of opposite side

45. In any triangle, a circle can be drawn tangent to the three sides.

• This is true for the incircle of a triangle, which is tangent to all three sides.
• Answer: C. True

46. In any triangle, a circle can be drawn that will pass through the three vertices.

• This is true for the circumcircle of a triangle, where the circle passes through all three vertices.
• Answer: B. True

47. The centroid of a triangle always passes through its:

• The centroid is the point of intersection of the medians of a triangle.
• Answer: A. Median

Would you like more detailed explanations on any of these questions?

Related Questions:

1. How do you derive the formula for the area of a square?
2. What are the properties of a regular polygon with respect to the apothem?
3. How do you calculate the apothem for other polygons (besides a hexagon)?
4. What is the difference between the incircle and circumcircle in a triangle?
5. Can the centroid of a triangle be outside of the triangle?

Tip:

In any regular polygon, the apothem can be found using both the perimeter and area. It is essential to use this when calculating areas more efficiently.