Question

determine the perimeter of the given figure on the coordinate plane. round your answer to the nearest hundredth, if necessary. triangle def

Explanation:

1. First, assume the coordinates of the vertices of the triangle:

• Let's assume from the graph that \(D(-6,-5)\), \(E(-4,1)\), and \(F(2, - 5)\).
• The distance formula between two - points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

1. Calculate the length of side \(DE\):

• Here, \(x_1=-6,y_1 = - 5,x_2=-4,y_2 = 1\).
• \(d_{DE}=\sqrt{(-4+6)^2+(1 + 5)^2}=\sqrt{(2)^2+(6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\approx6.32\).

1. Calculate the length of side \(EF\):

• Here, \(x_1=-4,y_1 = 1,x_2=2,y_2=-5\).
• \(d_{EF}=\sqrt{(2 + 4)^2+(-5 - 1)^2}=\sqrt{(6)^2+(-6)^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}\approx8.49\).

1. Calculate the length of side \(DF\):

• Here, \(x_1=-6,y_1=-5,x_2=2,y_2=-5\).
• Since \(y_1 = y_2=-5\), the distance \(d_{DF}=\vert2+6\vert = 8\).

1. Calculate the perimeter \(P\) of \(\triangle DEF\):

• \(P=d_{DE}+d_{EF}+d_{DF}\).
• \(P\approx6.32 + 8.49+8\).
• \(P\approx22.81\).

Step1: Identify vertex coordinates

Assume \(D(-6,-5)\), \(E(-4,1)\), \(F(2,-5)\).

Step2: Calculate length of \(DE\)

Use distance - formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), get \(d_{DE}=\sqrt{(-4 + 6)^2+(1 + 5)^2}=\sqrt{40}\approx6.32\).

Step3: Calculate length of \(EF\)

Use distance - formula, get \(d_{EF}=\sqrt{(2 + 4)^2+(-5 - 1)^2}=\sqrt{72}\approx8.49\).

Step4: Calculate length of \(DF\)

Since \(y\) - coordinates are the same, \(d_{DF}=\vert2+6\vert = 8\).

Step5: Calculate perimeter

\(P=d_{DE}+d_{EF}+d_{DF}\approx6.32+8.49 + 8=22.81\).

Answer:

\(22.81\)