example 1 find the perimeter or circumference and area of each figure if each unit on the graph measures 1 centimeter. round answers to the nearest tenth, if necessary. 1. a(0, 5) b(4, 9) c(0, 1) 2. a(-2, 7) b(4, 4) d(-4, 3) c(2, 0) 3. a(0, 3) b(4, 1) 4. x(-3, 4) y(-4, 2) 5. g(3, -1) h(1, -3) f(5, -3) e(3, -5) 6. a(0, 4) b(2, 0) c(-4, 0) example 2 use a two - dimensional model and the dimensions provided to calculate the perimeter or circumference and area of each object. round to the nearest tenth, if necessary. 7. 5 in. 8. 14.87 yd 11.92 yd 13.44 yd 9. 6.25 ft

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1. **For the triangle in problem 1 with vertices $A(0,5)$, $B(4,9)$, $C(0,1)$**:
   - **Find the lengths of the sides using the distance - formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$**:
     - **Step 1: Find the length of $AB$**
       - Let $(x_1,y_1)=(0,5)$ and $(x_2,y_2)=(4,9)$. Then $AB=\sqrt{(4 - 0)^2+(9 - 5)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.7$ cm.
     - **Step 2: Find the length of $BC$**
       - Let $(x_1,y_1)=(4,9)$ and $(x_2,y_2)=(0,1)$. Then $BC=\sqrt{(0 - 4)^2+(1 - 9)^2}=\sqrt{16 + 64}=\sqrt{80}\approx8.9$ cm.
     - **Step 3: Find the length of $AC$**
       - Let $(x_1,y_1)=(0,5)$ and $(x_2,y_2)=(0,1)$. Then $AC=\sqrt{(0 - 0)^2+(1 - 5)^2}=\sqrt{0 + 16}=4$ cm.
     - **Step 4: Calculate the perimeter $P$**
       - $P=AB + BC+AC\approx5.7+8.9 + 4=18.6$ cm.
     - **Step 5: Calculate the area $A$ using the formula for the area of a triangle $A=\frac{1}{2}\times base\times height$**
       - The base $AC = 4$ cm and the height (the horizontal distance from $B$ to the line $x = 0$) is $4$ cm. So $A=\frac{1}{2}\times4\times4 = 8$ $cm^2$.
2. **For the parallelogram in problem 2 with vertices $A(-2,7)$, $B(4,4)$, $C(2,0)$, $D(-4,3)$**:
   - **Find the lengths of two adjacent sides using the distance - formula**:
     - **Step 1: Find the length of $AB$**
       - Let $(x_1,y_1)=(-2,7)$ and $(x_2,y_2)=(4,4)$. Then $AB=\sqrt{(4+2)^2+(4 - 7)^2}=\sqrt{36 + 9}=\sqrt{45}\approx6.7$ cm.
     - **Step 2: Find the length of $BC$**
       - Let $(x_1,y_1)=(4,4)$ and $(x_2,y_2)=(2,0)$. Then $BC=\sqrt{(2 - 4)^2+(0 - 4)^2}=\sqrt{4 + 16}=\sqrt{20}\approx4.5$ cm.
     - **Step 3: Calculate the perimeter $P$**
       - Since opposite sides of a parallelogram are equal, $P = 2(AB + BC)\approx2(6.7+4.5)=22.4$ cm.
     - **Step 4: Calculate the area $A$ using the cross - product method (or by finding the height and base)**
       - We can use the formula for the area of a parallelogram given two adjacent vectors $\vec{u}$ and $\vec{v}$. Another way is to find the base and height. If we consider the base as the distance between two points on a horizontal or vertical line. Let's use the fact that the area of a parallelogram $A = base\times height$. The base $AB\approx6.7$ cm and the height (the perpendicular distance from $C$ to $AB$) can be found using the formula for the distance between a point and a line. First, find the equation of the line $AB$: The slope $m=\frac{4 - 7}{4+2}=-\frac{1}{2}$, and using the point - slope form with point $A(-2,7)$, $y - 7=-\frac{1}{2}(x + 2)$ or $x+2y=12$. The distance from point $C(2,0)$ to the line $x + 2y-12=0$ is $d=\frac{|2+2\times0 - 12|}{\sqrt{1^2+2^2}}=\frac{10}{\sqrt{5}}\approx4.5$ cm. So $A=AB\times h\approx6.7\times4.5 = 30.2$ $cm^2$.
3. **For the circle in problem 3 with center $A(0,3)$ and a point $B(4,1)$ on the circle**:
   - **First, find the radius $r$ using the distance formula**:
     - **Step 1: Calculate the radius $r$**
       - Let $(x_1,y_1)=(0,3)$ and $(x_2,y_2)=(4,1)$. Then $r=\sqrt{(4 - 0)^2+(1 - 3)^2}=\sqrt{16 + 4}=\sqrt{20}\approx4.5$ cm.
     - **Step 2: Calculate the circumference $C$**
       - Using the formula $C = 2\pi r$, $C=2\pi\times4.5\approx28.3$ cm.
     - **Step 3: Calculate the area $A$**
       - Using the formula $A=\pi r^2$, $A=\pi\times(4.5)^2\approx63.6$ $cm^2$.
4. **For the circle in problem 4 with center $X(-3,4)$ and a point $Y(-4,2)$ on the circle**:
   - **Find the radius $r$ using the distance formula**:
     - **Step 1: Calculate the radius $r$**
       - Let $(x_1,y_1)=(-3,4)$ and $(x_2,y_2)=(-4,2)$. Then $r=\sqrt{(-4 + 3)^2+(2 - 4)^2}=\sqrt{1+4}=\sqrt{5}\approx2.2$ cm.
     - **Step 2: Calculate the circumference $C$**
       - $C = 2\pi r=2\pi\times2.2\approx13.8$ cm.
     - **Step 3: Calculate the area $A$**
       - $A=\pi r^2=\pi\times(2.2)^2\approx15.2$ $cm^2$.
5. **For the rhombus in problem 5 with vertices $E(3,-5)$, $F(5,-3)$, $G(3,-1)$, $H(1,-3)$**:
   - **Find the length of a side using the distance formula**:
     - **Step 1: Find the length of a side, say $EF$**
       - Let $(x_1,y_1)=(3,-5)$ and $(x_2,y_2)=(5,-3)$. Then $EF=\sqrt{(5 - 3)^2+(-3 + 5)^2}=\sqrt{4 + 4}=\sqrt{8}\approx2.8$ cm.
     - **Step 2: Calculate the perimeter $P$**
       - Since all sides of a rhombus are equal, $P = 4EF\approx4\times2.8 = 11.2$ cm.
     - **Step 3: Calculate the area $A$ using the formula $A=\frac{1}{2}d_1d_2$ (where $d_1$ and $d_2$ are the lengths of the diagonals)**
       - The diagonals: The mid - points of the diagonals of a rhombus bisect each other. The length of one diagonal (the vertical distance between the mid - points of the horizontal lines connecting the vertices) $d_1 = 4$ cm and the other diagonal (the horizontal distance between the mid - points of the vertical lines connecting the vertices) $d_2 = 4$ cm. So $A=\frac{1}{2}\times4\times4 = 8$ $cm^2$.
6. **For the triangle in problem 6 with vertices $A(0,4)$, $B(2,0)$, $C(-4,0)$**:
   - **Find the lengths of the sides using the distance formula**:
     - **Step 1: Find the length of $AB$**
       - Let $(x_1,y_1)=(0,4)$ and $(x_2,y_2)=(2,0)$. Then $AB=\sqrt{(2 - 0)^2+(0 - 4)^2}=\sqrt{4 + 16}=\sqrt{20}\approx4.5$ cm.
     - **Step 2: Find the length of $BC$**
       - Let $(x_1,y_1)=(2,0)$ and $(x_2,y_2)=(-4,0)$. Then $BC=\sqrt{(-4 - 2)^2+(0 - 0)^2}=6$ cm.
     - **Step 3: Find the length of $AC$**
       - Let $(x_1,y_1)=(0,4)$ and $(x_2,y_2)=(-4,0)$. Then $AC=\sqrt{(-4 - 0)^2+(0 - 4)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.7$ cm.
     - **Step 4: Calculate the perimeter $P$**
       - $P=AB + BC+AC\approx4.5+6 + 5.7=16.2$ cm.
     - **Step 5: Calculate the area $A$ using the formula $A=\frac{1}{2}\times base\times height$**
       - The base $BC = 6$ cm and the height (the vertical distance from $A$ to the line $y = 0$) is $4$ cm. So $A=\frac{1}{2}\times6\times4 = 12$ $cm^2$.
7. **For the circle in problem 7 with diameter $d = 5$ in**:
   - **Step 1: Calculate the radius $r=\frac{d}{2}=2.5$ in**
   - **Step 2: Calculate the circumference $C$**
     - Using the formula $C=\pi d$, $C = 5\pi\approx15.7$ in.
   - **Step 3: Calculate the area $A$**
     - Using the formula $A=\pi r^2=\pi\times(2.5)^2\approx19.6$ $in^2$.
8. **For the triangle in problem 8 with side lengths $a = 14.87$ yd, $b = 13.84$ yd, $c = 11.92$ yd**:
   - **Step 1: Calculate the perimeter $P$**
     - $P=a + b + c=14.87+13.84+11.92=40.63\approx40.6$ yd.
     - **Step 2: Calculate the area $A$ using Heron's formula. First, find the semi - perimeter $s=\frac{a + b + c}{2}=\frac{40.63}{2}=20.315$ yd**
     - $A=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{20.315(20.315 - 14.87)(20.315 - 13.84)(20.315 - 11.92)}\approx79.9$ $yd^2$.
9. **For the rectangle (assuming the van's shape is approximated as a rectangle) in problem 9 with length $l = 14.8$ ft and width $w = 6.25$ ft**:
   - **Step 1: Calculate the perimeter $P$**
     - Using the formula $P = 2(l + w)$, $P=2(14.8+6.25)=42.1$ ft.
   - **Step 2: Calculate the area $A$**
     - Using the formula $A=lw$, $A=14.8\times6.25 = 92.5$ $ft^2$.

# Answer:
1. Perimeter: $18.6$ cm, Area: $8$ $cm^2$
2. Perimeter: $22.4$ cm, Area: $30.2$ $cm^2$
3. Circumference: $28.3$ cm, Area: $63.6$ $cm^2$
4. Circumference: $13.8$ cm, Area: $15.2$ $cm^2$
5. Perimeter: $11.2$ cm, Area: $8$ $cm^2$
6. Perimeter: $16.2$ cm, Area: $12$ $cm^2$
7. Circumference: $15.7$ in, Area: $19.6$ $in^2$
8. Perimeter: $40.6$ yd, Area: $79.9$ $yd^2$
9. Perimeter: $42.1$ ft, Area: $92.5$ $ft^2$

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