https://file.aiquickdraw.com/mathbot/file/1730753039939xma7f2n1.jpg

Let's go through each part of the question step by step.

### Part (a)
**Explanation for why triangle \( DEC \) is equilateral:**

Since \( ABCD \) is a square with a side length of \( 62 \) mm:
1. The sides \( AD \), \( DC \), \( AB \), and \( BC \) are all equal, so each has a length of \( 62 \) mm.
2. Arc \( AC \) is drawn with center \( D \), meaning the radius of this arc is \( DA = DC = 62 \) mm.
3. Similarly, arc \( BD \) is drawn with center \( C \), making \( CB = CD = 62 \) mm.

In triangle \( DEC \):
- \( DE = DC = CE = 62 \) mm, so all three sides are equal.
  
Since all sides of \( DEC \) are equal, triangle \( DEC \) is an **equilateral triangle**.

### Part (b)
**Calculating the perimeter of the shaded shape \( ABE \):**

The shaded shape \( ABE \) consists of:
1. Line segment \( AB \).
2. Arc \( AE \) (a part of arc \( AC \)).
3. Arc \( BE \) (a part of arc \( BD \)).

Given:
- \( AB = 62 \) mm (since it's a side of the square).
- Both arcs \( AE \) and \( BE \) subtend an angle of \( 60^\circ \) (since they each span one side of the equilateral triangle \( DEC \)).

To find the arc lengths:
1. **Arc \( AE \):** 
   - Arc \( AE \) is \( \frac{60^\circ}{360^\circ} \times \) the circumference of the circle centered at \( D \) with radius \( 62 \) mm.
   - Circumference of the circle = \( 2 \pi \times 62 = 124 \pi \).
   - Arc \( AE = \frac{60}{360} \times 124 \pi = \frac{1}{6} \times 124 \pi = \frac{124 \pi}{6} \).

2. **Arc \( BE \):**
   - Similarly, arc \( BE = \frac{124 \pi}{6} \).

Now, adding up these lengths for the perimeter of \( ABE \):
\[
\text{Perimeter of } ABE = AB + \text{Arc } AE + \text{Arc } BE
\]
\[
= 62 + \frac{124 \pi}{6} + \frac{124 \pi}{6}
\]
\[
= 62 + \frac{248 \pi}{6}
\]
\[
= 62 + \frac{124 \pi}{3}
\]

Using \( \pi \approx 3.1416 \):
\[
\text{Perimeter} \approx 62 + \frac{124 \times 3.1416}{3}
\]
\[
\approx 62 + 129.9467
\]
\[
\approx 191.9 \text{ mm (to 1 decimal place)}
\]

### Final Answer:
- **(a)** Triangle \( DEC \) is equilateral because all three sides are equal in length (62 mm).
- **(b)** The perimeter of the shaded shape \( ABE \) is approximately **191.9 mm**.

Would you like further details or clarification on any part?

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### Related Questions:
1. How would the perimeter change if the side length of the square was different?
2. What would be the area of the shaded region \( ABE \)?
3. How would you calculate the perimeter if \( ABE \) included a different arc length?
4. Can you find the area of triangle \( DEC \)?
5. What if \( AC \) and \( BD \) were semicircles instead of arcs of \( 60^\circ \)?

### Tip:
When dealing with arc lengths, always make sure to convert angles to fractions of \( 360^\circ \) to find their respective portions of the circle’s circumference.

ABCD is a square with a side length of 62 mm. AC is an arc of a circle with center D. BD is an arc of a circle with center C. a) Write a sentence to explain how you know triangle DEC is equilateral. b) Calculate the perimeter of the shaded shape ABE to 1 d.p.

This problem involves calculating the perimeter of a shaded shape in a square with given side length and arc radii. Key concepts include equilateral triangles, arc length calculation, and perimeter of composite shapes. Suitable for students in Grades 9-11.

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