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To solve this problem, we need to work with both the triangle \(ABE\) and the sector \(BCDE\), and calculate the area and perimeter of the combined shape.

### Given Data:
- \( AB = 19 \, \text{cm} \)
- \( BC = 8 \, \text{cm} \)
- Radius of the sector \( BCDE = 8 \, \text{cm} \)
- Angle \( ABE = 0.73 \, \text{radians} \)

We will solve this step-by-step.

### Part (a) The area of the shape:

1. **Area of the triangle \(ABE\):**
   The area \(A_{\triangle}\) of a triangle can be calculated using:
   \[
   A_{\triangle} = \frac{1}{2} \cdot AB \cdot BE \cdot \sin(\theta)
   \]
   where \(AB = 19 \, \text{cm}\), \(BE = 8 \, \text{cm}\), and \( \theta = 0.73 \, \text{radians} \).

2. **Area of the sector \(BCDE\):**
   The area \(A_{\text{sector}}\) of a sector of a circle is given by:
   \[
   A_{\text{sector}} = \frac{1}{2} \cdot r^2 \cdot \theta
   \]
   where \(r = 8 \, \text{cm}\) and \( \theta = 0.73 \, \text{radians} \).

We will now compute both areas and add them up.

### Part (b) The perimeter of the shape:

1. **Perimeter of the triangle \(ABE\):**
   The perimeter is the sum of the sides \(AB\), \(BE\), and the arc length \(AE\).

2. **Arc length of sector \(BCDE\):**
   The arc length \(L\) of a sector is given by:
   \[
   L = r \cdot \theta
   \]
   where \(r = 8 \, \text{cm}\) and \( \theta = 0.73 \, \text{radians} \).

After calculating, we will sum these lengths to get the total perimeter of the shape.

Let's calculate these now.### Results:

1. **Area of the shape**: \( 74.0 \, \text{cm}^2 \) (to 3 significant figures)
2. **Perimeter of the shape**: \( 32.8 \, \text{cm} \) (to 3 significant figures)

Would you like more details or clarifications on any part of the solution?

Here are some related questions to consider:
1. How would the area change if the angle \( ABE \) were increased?
2. What formula is used for the perimeter of more complex composite shapes?
3. How is the arc length related to the radius and angle in radians?
4. Can the sine function be used to calculate triangle areas with any given angle unit?
5. How does the formula for the area of a sector differ from that of a circle?

**Tip:** Always ensure angles are in the correct units (degrees or radians) when applying trigonometric functions or calculating areas/lengths involving sectors.

The shape consists of a triangle ABE joined to a sector BCDE of a circle with radius 8cm and center B. The points A, B, and C lie on a straight line with AB = 19cm and BC = 8cm. Given that the size of angle ABE is exactly 0.73 radians, find, giving your answers to 3 significant figures: a) the area of the shape, b) the perimeter of the shape.

This geometry problem involves finding the area and perimeter of a shape combining a triangle ABE and a sector BCDE. Key data includes AB = 19cm, BC = 8cm, and the angle ABE = 0.73 radians. Learn how to use trigonometric and geometric formulas to solve for the area and perimeter of the shape.

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