Question

find the perimeter of the figure. round to the nearest tenth of a unit. 9. 10. 11.

Explanation:

Let's solve problem 9 first. We need to find the perimeter of quadrilateral \(ABCD\) by calculating the lengths of its sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 1: Identify coordinates of points

From the grid:

• \(A(1, 1)\)
• \(B(2, -1)\)
• \(C(5, 1)\)
• \(D(4, 3)\)

Step 2: Calculate length of \(AB\)

Using distance formula between \(A(1, 1)\) and \(B(2, -1)\):
\[
AB = \sqrt{(2 - 1)^2 + (-1 - 1)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

Step 3: Calculate length of \(BC\)

Between \(B(2, -1)\) and \(C(5, 1)\):
\[
BC = \sqrt{(5 - 2)^2 + (1 - (-1))^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.6
\]

Step 4: Calculate length of \(CD\)

Between \(C(5, 1)\) and \(D(4, 3)\):
\[
CD = \sqrt{(4 - 5)^2 + (3 - 1)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

Step 5: Calculate length of \(DA\)

Between \(D(4, 3)\) and \(A(1, 1)\):
\[
DA = \sqrt{(1 - 4)^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.6
\]

Step 6: Calculate perimeter

Perimeter \(= AB + BC + CD + DA\)
\[
\text{Perimeter} \approx 2.2 + 3.6 + 2.2 + 3.6 = 11.6
\]

For problem 10 (quadrilateral \(URST\)):

Step 1: Identify coordinates

• \(U(0, 0)\)
• \(R(2, 3)\)
• \(S(4, 1)\)
• \(T(3, -1)\)

Step 2: Calculate \(UR\)

\[
UR = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6
\]

Step 3: Calculate \(RS\)

\[
RS = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8
\]

Step 4: Calculate \(ST\)

\[
ST = \sqrt{(3 - 4)^2 + (-1 - 1)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

Step 5: Calculate \(TU\)

\[
TU = \sqrt{(0 - 3)^2 + (0 - (-1))^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.2
\]

Step 6: Perimeter

\[
\text{Perimeter} \approx 3.6 + 2.8 + 2.2 + 3.2 = 11.8
\]

For problem 11 (triangle \(EFG\)):

Step 1: Identify coordinates

• \(E(-2, 3)\)
• \(F(2, 3)\)
• \(G(0, -2)\)

Step 2: Calculate \(EF\)

\[
EF = \sqrt{(2 - (-2))^2 + (3 - 3)^2} = \sqrt{16 + 0} = 4
\]

Step 3: Calculate \(EG\)

\[
EG = \sqrt{(0 - (-2))^2 + (-2 - 3)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.4
\]

Step 4: Calculate \(FG\)

\[
FG = \sqrt{(0 - 2)^2 + (-2 - 3)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.4
\]

Step 5: Perimeter

\[
\text{Perimeter} \approx 4 + 5.4 + 5.4 = 14.8
\]

Final Answers:

1. \(\boxed{11.6}\)
2. \(\boxed{11.8}\)
3. \(\boxed{14.8}\)

Answer:

Let's solve problem 9 first. We need to find the perimeter of quadrilateral \(ABCD\) by calculating the lengths of its sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 1: Identify coordinates of points

From the grid:

• \(A(1, 1)\)
• \(B(2, -1)\)
• \(C(5, 1)\)
• \(D(4, 3)\)

Step 2: Calculate length of \(AB\)

Using distance formula between \(A(1, 1)\) and \(B(2, -1)\):
\[
AB = \sqrt{(2 - 1)^2 + (-1 - 1)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

Step 3: Calculate length of \(BC\)

Between \(B(2, -1)\) and \(C(5, 1)\):
\[
BC = \sqrt{(5 - 2)^2 + (1 - (-1))^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.6
\]

Step 4: Calculate length of \(CD\)

Between \(C(5, 1)\) and \(D(4, 3)\):
\[
CD = \sqrt{(4 - 5)^2 + (3 - 1)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

Step 5: Calculate length of \(DA\)

Between \(D(4, 3)\) and \(A(1, 1)\):
\[
DA = \sqrt{(1 - 4)^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.6
\]

Step 6: Calculate perimeter

Perimeter \(= AB + BC + CD + DA\)
\[
\text{Perimeter} \approx 2.2 + 3.6 + 2.2 + 3.6 = 11.6
\]

For problem 10 (quadrilateral \(URST\)):

Step 1: Identify coordinates

• \(U(0, 0)\)
• \(R(2, 3)\)
• \(S(4, 1)\)
• \(T(3, -1)\)

Step 2: Calculate \(UR\)

\[
UR = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6
\]

Step 3: Calculate \(RS\)

\[
RS = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8
\]

Step 4: Calculate \(ST\)

\[
ST = \sqrt{(3 - 4)^2 + (-1 - 1)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

Step 5: Calculate \(TU\)

\[
TU = \sqrt{(0 - 3)^2 + (0 - (-1))^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.2
\]

Step 6: Perimeter

\[
\text{Perimeter} \approx 3.6 + 2.8 + 2.2 + 3.2 = 11.8
\]

For problem 11 (triangle \(EFG\)):

Step 1: Identify coordinates

• \(E(-2, 3)\)
• \(F(2, 3)\)
• \(G(0, -2)\)

Step 2: Calculate \(EF\)

\[
EF = \sqrt{(2 - (-2))^2 + (3 - 3)^2} = \sqrt{16 + 0} = 4
\]

Step 3: Calculate \(EG\)

\[
EG = \sqrt{(0 - (-2))^2 + (-2 - 3)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.4
\]

Step 4: Calculate \(FG\)

\[
FG = \sqrt{(0 - 2)^2 + (-2 - 3)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.4
\]

Step 5: Perimeter

\[
\text{Perimeter} \approx 4 + 5.4 + 5.4 = 14.8
\]

Final Answers:

1. \(\boxed{11.6}\)
2. \(\boxed{11.8}\)
3. \(\boxed{14.8}\)