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### Part a) Write a fully factorised expression for the perimeter of the kite.

To find the perimeter of the kite, we add all four sides together. From the diagram:

- The two shorter sides are \( 6(x + 3y) \) and \( 6(x + 3y) \).
- The two longer sides are \( 9(2x - y) \) and \( 9(2x - y) \).

The total perimeter \( P \) is:
\[
P = 2 \times 6(x + 3y) + 2 \times 9(2x - y)
\]
\[
P = 12(x + 3y) + 18(2x - y)
\]
Now, expand each term:
\[
P = 12x + 36y + 36x - 18y
\]
Combine like terms:
\[
P = (12x + 36x) + (36y - 18y)
\]
\[
P = 48x + 18y
\]
Now, factor the expression:
\[
P = 6(8x + 3y)
\]

So, the fully factorised expression for the perimeter of the kite is:
\[
6(8x + 3y)
\]

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### Part b) How many times larger than the circumference of the circle is the perimeter of the kite?

The circumference of the circle is given by:
\[
\text{Circumference} = 6(x + y) + 10x
\]
Simplify this expression:
\[
\text{Circumference} = 6x + 6y + 10x
\]
\[
\text{Circumference} = 16x + 6y
\]

Now, to find how many times larger the perimeter of the kite is compared to the circumference of the circle, we divide the perimeter of the kite by the circumference of the circle:
\[
\frac{\text{Perimeter of kite}}{\text{Circumference of circle}} = \frac{6(8x + 3y)}{16x + 6y}
\]
We can factor out 2 from the denominator:
\[
\frac{6(8x + 3y)}{2(8x + 3y)} = 3
\]

So, the perimeter of the kite is **3 times larger** than the circumference of the circle.

---

Would you like more details or have any questions? Here are five related questions to explore:

1. How would the perimeter change if the dimensions of the kite sides were scaled by a factor of 2?
2. Can you find the area of the kite based on the given side lengths?
3. What if the circle's circumference formula were altered? How would that affect the ratio in part b?
4. How do we interpret the factorised expressions geometrically?
5. Can you generalize the relationship between perimeters and circumferences of different polygons and circles?

**Tip:** Always check for common factors when simplifying algebraic expressions—this helps in quickly finding factored forms.

A kite and a circle are shown below. a) Write a fully factorised expression for the perimeter of the kite. b) How many times larger than the circumference of the circle is the perimeter of the kite?

This problem focuses on finding the fully factorised expression for the perimeter of a kite and comparing it with the circumference of a circle. Involves key algebraic skills like factorization and the application of geometry. Suitable for Grades 8-10.

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