Question: 20. a) Write an equation for the circle in the diagram below.
b) Write the coordinates of points B and D.
c) Determine equations for the lines containing line segments AB and CD.
d) Determine the coordinates of point E.
e) Determine the area of the shaded portion of the circle.
Graph: A coordinate plane with a circle centered at the origin; the circle has radius 4 and passes through top point C at (0,4), left point near B at (-4,0), and bottom point near (0,-4). A slanted line segment/chord from C to A slopes downward to the right and another slanted line from B through E to A slopes downward to the right; the shaded region is the circular segment in the upper-right part of the circle between the two chords. position_hint = centerUser: Solve for e)
1. **Problem:** Determine the area of the shaded portion of the circle.
2. **Formula used:** For a circular sector, the area is
$$A_{\text{sector}}=\frac{\theta}{360^\circ}\pi r^2$$
and for a triangle with two sides $r$ and included angle $\theta$,
$$A_{\triangle}=\frac{1}{2}r^2\sin(\theta)$$
The shaded region is the **sector minus the triangle**.
3. **Important rule:** From the diagram, the circle is centered at the origin and has radius $4$. The shaded part is the region cut off by the two radii to the endpoints, so it is a circular segment.
4. **Find the central angle:** The endpoints of the shaded arc are at the top point $C(0,4)$ and the right point $A(4,0)$. These points are $90^\circ$ apart from the center, so
$$\theta=90^\circ$$
5. **Find the sector area:**
$$A_{\text{sector}}=\frac{90^\circ}{360^\circ}\pi(4)^2$$
$$A_{\text{sector}}=\frac{1}{4}\pi\cdot 16$$
$$A_{\text{sector}}=4\pi$$
6. **Find the triangle area:**
$$A_{\triangle}=\frac{1}{2}(4)(4)\sin(90^\circ)$$
$$A_{\triangle}=\frac{1}{2}\cdot 16\cdot 1$$
$$A_{\triangle}=8$$
7. **Subtract to get the shaded area:**
$$A_{\text{shaded}}=A_{\text{sector}}-A_{\triangle}$$
$$A_{\text{shaded}}=4\pi-8$$
8. **Final answer:** The area of the shaded portion is
$$\boxed{4\pi-8}$$