Multiple geometry questions.

Steps to Solve

Here's the breakdown of the True/False and Fill in the Blanks questions with their solutions:

II. TRUE / FALSE

1. Electric Poles: $192 / 8 = 24$. This statement is true.

2. Area of a circle: The area of a circle is indeed $\pi r^2$. This statement is true.

3. Area of a triangle: The area of a triangle is $\frac{1}{2} \times \text{Base} \times \text{Height}$, not just Base x Height. This statement is false.

4. Plants around a circle: The circumference of the circle is $2 \pi r = 2 \times \pi \times 42 = 84\pi \approx 84 \times \frac{22}{7} = 12 \times 22 = 264$. Number of plants $= 264 / 4 = 66$. This statement is true.

5. Side of a parallelogram: Area of parallelogram $=$ base $\times$ height. So, base $= \frac{\text{area of parallelogram}}{\text{height}}$. Side of parallelogram can be the base. This statement is true.

6. Circle radius and circumference: If the radius is 7 cm, the circumference is $2 \pi r = 2 \times \pi \times 7 = 14\pi \approx 14 \times \frac{22}{7} = 2 \times 22 = 44$ cm, not 48 cm. This statement is false.

7. Area of a square: A square with a side of 1 cm has an area of $1 , \text{cm}^2$. $1 , \text{m}^2$ is equal to $10000 , \text{cm}^2$. Thus, the statement is false.

8. Area of a rhombus: Area of rhombus $= \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 24 \times 18 = 12 \times 18 = 216 , \text{cm}^2$. This statement is true.

9. Perimeter and Area: Consider a square with side $x$. Its perimeter is $4x$ and its area is $x^2$. Now consider a rectangle with sides $x-a$ and $x+a$. The perimeter if $2(x-a)+2(x+a) = 4x$. The area is $(x-a)(x+a) = x^2 - a^2$, Since $a^2 > 0$, $x^2 > x^2 - a^2$ so $A_1 > A_2$. Therefore this statement: $A_2 > A_1$ is false.

10. Perimeter definition: Perimeter is the distance around a closed figure. This statement is true.

III. FILL IN THE BLANKS

1. Area of triangle: Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height}$. So the blank should be "height".

2. Circle circumscribed about a square: Side of square $= a$. Diagonal of square $= a\sqrt{2}$. Radius of the circle $= \frac{a\sqrt{2}}{2}$. Area of circle $= \pi r^2 = \pi (\frac{a\sqrt{2}}{2})^2 = \pi \frac{a^2 \times 2}{4} = \frac{\pi a^2}{2}$.

3. Measure of the region: The measure of the region enclosed by a plane figure is called its area.

4. Area in terms of r and C: Area, $A = \pi r^2$. Circumference, $C = 2 \pi r$, so $r = \frac{C}{2\pi}$. Therefore, $A = \pi (\frac{C}{2\pi})^2 = \pi \frac{C^2}{4 \pi^2} = \frac{C^2}{4\pi} = \frac{1}{2} * C * r$.

5. Cost of putting a lane: To find the cost of putting a lane around a circular table cover, we find the circumference.

6. Distance around a circular region: The distance around a circular region is called its circumference.

7. Perimeter of an octagon: The length of a regular octagon is 4 cm, then its perimeter is $8 \times 4 = 32$ cm.

8. Area of a circular ring: The area of circular ring with outer and inner radii $r_1$ and $r_2$ respectively is $\pi (r_1^2 - r_2^2)$.

9. Side of a square is doubled: If the side of a square is doubled, then its perimeter becomes 2 times the original perimeter.

10. Height and base of triangle halved: If both the height and base of a triangle are halved, then its area becomes $\frac{1}{4}$ times the original area.