Question

choose 1 answer:
a 30
b $16 + 10\sqrt{17}$
c $16 + 2\sqrt{85}$
d $6 + 2\sqrt{7} + 2\sqrt{13}$

Explanation:

Step1: Calculate the length of \(DC\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(D(-3,-4)\) and \(C(3,-4)\), \(x_1=-3,y_1 = - 4,x_2=3,y_2=-4\).

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Since \(ED = BC\) and \(AE=AB\), \(ED = BC = 6\). So \(DC+ED+BC=6 + 6+6=18\) (This is wrong, we should calculate \(DC = 6\), and then calculate the length of \(AE\) (or \(AB\)) and \(ED\) (or \(BC\)) correctly. Let's start over)

Step1: Calculate the length of \(DC\)

For \(D(-3,-4)\) and \(C(3,-4)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(x_1=-3,y_1=-4,x_2 = 3,y_2=-4\).

$$DC=\sqrt{(3-(-3))^2+(-4-(-4))^2}=\sqrt{6^2+0^2}=6$$

Step2: Calculate the length of \(AE\)

For \(A(0,7)\) and \(E(-6,0)\), \(x_1 = 0,y_1=7,x_2=-6,y_2 = 0\)

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Since \(AE = AB\) and \(ED=BC\). \(ED=\sqrt{( - 3+6)^2+(-4 - 0)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=5\), \(BC=\sqrt{(6 - 3)^2+(0+4)^2}=\sqrt{3^2+4^2}=5\)
The perimeter \(P=DC + CB+BA+AE+ED\). \(DC = 6\), \(CB = 5\), \(BA=\sqrt{(6-0)^2+(0 - 7)^2}=\sqrt{36+49}=\sqrt{85}\), \(AE=\sqrt{85}\), \(ED = 5\)

$$P=6+5+\sqrt{85}+\sqrt{85}+5=16 + 2\sqrt{85}$$

Answer:

C. \(16 + 2\sqrt{85}\)