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Let's go through the problems one by one:

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**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**

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**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**

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**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**

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**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**

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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.

41. The perimeter of a square is 70 cm. Its area (cm²) is: A. 654.75, B. 306.25, C. 4900, D. 1225
42. What is the apothem of a regular polygon having an area of 225 sq.cm. and a perimeter of 60 cm? A. 4.5, B. 5.5, C. 6.5, D. 7.5
43. Find the area of the inscribed circle of a regular hexagon whose side has length 8. A. 50π, B. 48π, C. 60π, D. 80π
44. The median of a triangle is a line drawn from a vertex: A. none of these, B. dividing the triangle into two equal areas, C. bisecting the angle to that vertex, D. to the midpoint of opposite side
45. In any triangle, a circle can be drawn tangent to the three sides. A. false, B. it depends, C. true, D. none of these
46. In any triangle, a circle can be drawn that will pass through the three vertices. A. it depends, B. true, C. it depends, D. false
47. The centroid of a triangle always passes through its: A. median, B. altitude, C. angle bisector, D. side

Solve a series of geometry problems involving the perimeter and area of squares, the apothem of a regular polygon, the area of an inscribed circle of a hexagon, and properties of medians, centroids, and incircles of triangles. Suitable for students in Grades 9-12.

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