Question

1. for problems 2-5, sketch a polygon inside the circle and estimate the area of the circle. explain your thinking.

Explanation:

Step1: Identify circle parameters

The circle has center at $(4,5)$ and radius $r=3$ (since it spans 3 units from center to edge along x/y axes).

Step2: Inscribe square in circle

The inscribed square's diagonal equals $2r=6$. For a square, diagonal $d = s\sqrt{2}$, so side length $s=\frac{6}{\sqrt{2}}=3\sqrt{2}$.

Step3: Calculate square area

Area of square: $A_{square}=s^2=(3\sqrt{2})^2=18$.

Step4: Estimate circle area

A circle's area is ~1.57 times the inscribed square's area. Alternatively, use the circle area formula directly: $A_{circle}=\pi r^2$.
<Expression>
$A_{circle}=\pi (3)^2=9\pi\approx28.27$
</Expression>

Answer:

An inscribed square (with vertices at (1,5), (4,8), (7,5), (4,2)) can be drawn inside the circle. The estimated area of the circle is approximately 28 square units (or exactly $9\pi$ square units).