Let's analyze the mathematical problems from the image you uploaded.

1. First problem (i):

   • Question: "Find the perimeter of a regular hexagon inscribed in a circle whose radius is 100 m."
   • Solution:
     • A regular hexagon inscribed in a circle has all its vertices touching the circle. The distance from the center of the circle to any vertex is the radius of the circle, which is 100 meters.
     • Each side of the hexagon is equal to the radius of the circle, so the length of each side of the hexagon is 100 m.
     • The perimeter of the hexagon is given by: $\text{Perimeter} = 6 \times \text{side length}$ $\text{Perimeter} = 6 \times 100 = 600 \, \text{meters}.$
     • Answer: The perimeter is 600 m.

2. Second problem (ii):

   • Question: "Given that $\frac{AB}{KL} = \frac{BT}{LC} = \frac{TA}{CK} = 3$, where $AB$, $BT$, and $TA$ are sides of triangle $ABT$, and $KL$, $LC$, and $CK$ are the sides of triangle $KLC$, what does this information imply?"
   • Solution:
     • The equation $\frac{AB}{KL} = \frac{BT}{LC} = \frac{TA}{CK} = 3$ suggests that the corresponding sides of triangles $ABT$ and $KLC$ are proportional by a ratio of 3.
     • This indicates that triangle $ABT$ is similar to triangle $KLC$ by the Side-Side-Side (SSS) similarity criterion.
     • The constant ratio of 3 means that triangle $ABT$ is an enlargement of triangle $KLC$ by a factor of 3.
     • Answer: Triangle $ABT$ is similar to triangle $KLC$, with a scale factor of 3.

Would you like further details on these solutions, or do you have any specific questions?

Here are five related questions to expand on these problems:

1. What is the area of the regular hexagon inscribed in the circle?
2. How do you prove that two triangles are similar using the Side-Side-Side (SSS) criterion?
3. If the radius of the circle was doubled, what would be the new perimeter of the inscribed hexagon?
4. What are the properties of regular polygons inscribed in a circle?
5. How does the similarity ratio affect the area of similar triangles?

Tip: When dealing with regular polygons inscribed in circles, remember that the side length of the polygon is directly related to the radius of the circle.