Question

1. triangle abc

a(0,9) b(4,9)
c(0,1)
the area of the triangle is 16 square units

1. rectangle abcd

a(-2,7) b(4,4)
d(-4,3) c(2,0)
the perimeter of the rectangle is 22.36 units

1. circle a

a(2,3) b(4,1)
the circumference of the circle is units and the area is square units

Explanation:

Step1: Find the radius of the circle

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the distance between the center $A(2,3)$ and a point on the circle $B(4,1)$. Here $x_1 = 2,y_1 = 3,x_2=4,y_2 = 1$. So $r=\sqrt{(4 - 2)^2+(1 - 3)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.

Step2: Calculate the circumference of the circle

The formula for the circumference of a circle is $C = 2\pi r$. Substitute $r = 2\sqrt{2}$ into the formula, we get $C=2\pi\times2\sqrt{2}=4\sqrt{2}\pi$.

Step3: Calculate the area of the circle

The formula for the area of a circle is $A=\pi r^{2}$. Substitute $r = 2\sqrt{2}$ into the formula, we have $A=\pi\times(2\sqrt{2})^{2}=8\pi$.

Answer:

The circumference of the circle is $4\sqrt{2}\pi$ units and the area is $8\pi$ square units.