Question

type the correct answer in the box. use numerals instead of words.
for this item, any non - integer answer should be entered as a decimal, rounded to the hundredths place.
triangle abc has coordinates ( a(-4,1) ), ( b(0,-2) ), and ( c(4,1) ).
the perimeter of ( \triangle abc ) is units.

Explanation:

Step1: Recall the distance formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). We need to find the lengths of \(AB\), \(BC\), and \(AC\) and then sum them up to get the perimeter.

Step2: Calculate the length of \(AB\)

For points \(A(-4,1)\) and \(B(0,-2)\):
\(x_1=-4,y_1 = 1,x_2 = 0,y_2=-2\)
\(AB=\sqrt{(0 - (-4))^2+(-2 - 1)^2}=\sqrt{(4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\)

Step3: Calculate the length of \(BC\)

For points \(B(0,-2)\) and \(C(4,1)\):
\(x_1 = 0,y_1=-2,x_2 = 4,y_2 = 1\)
\(BC=\sqrt{(4 - 0)^2+(1 - (-2))^2}=\sqrt{(4)^2+(3)^2}=\sqrt{16+9}=\sqrt{25} = 5\)

Step4: Calculate the length of \(AC\)

For points \(A(-4,1)\) and \(C(4,1)\):
\(x_1=-4,y_1 = 1,x_2 = 4,y_2 = 1\)
\(AC=\sqrt{(4 - (-4))^2+(1 - 1)^2}=\sqrt{(8)^2+(0)^2}=\sqrt{64}=8\)

Step5: Calculate the perimeter

Perimeter \(P=AB + BC+AC\)
Substitute the values: \(P = 5+5 + 8=18\)

Answer:

18