To find the perimeter of rectangle ABCD with given vertices at coordinates [tex]\( A(-4, -2) \)[/tex], [tex]\( B(1, 10) \)[/tex], [tex]\( C(19, 2.5) \)[/tex], and [tex]\( D(14, -9.5) \)[/tex], we need to calculate the lengths of the four sides and then sum these lengths.
#### Step 1: Calculating the lengths of the sides
Let's determine the length of each side by calculating the distance between consecutive vertices. The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Length of side AB:
[tex]\[
A(-4, -2) \text{ to } B(1, 10)
\][/tex]
[tex]\[
AB = \sqrt{(1 - (-4))^2 + (10 - (-2))^2}
\][/tex]
[tex]\[
AB = \sqrt{(1 + 4)^2 + (10 + 2)^2}
\][/tex]
[tex]\[
AB = \sqrt{5^2 + 12^2}
\][/tex]
[tex]\[
AB = \sqrt{25 + 144}
\][/tex]
[tex]\[
AB = \sqrt{169} = 13 \text{ cm}
\][/tex]
2. Length of side BC:
[tex]\[
B(1, 10) \text{ to } C(19, 2.5)
\][/tex]
[tex]\[
BC = \sqrt{(19 - 1)^2 + (2.5 - 10)^2}
\][/tex]
[tex]\[
BC = \sqrt{18^2 + (-7.5)^2}
\][/tex]
[tex]\[
BC = \sqrt{324 + 56.25}
\][/tex]
[tex]\[
BC = \sqrt{380.25} = 19.5 \text{ cm}
\][/tex]
3. Length of side CD:
[tex]\[
C(19, 2.5) \text{ to } D(14, -9.5)
\][/tex]
[tex]\[
CD = \sqrt{(14 - 19)^2 + (-9.5 - 2.5)^2}
\][/tex]
[tex]\[
CD = \sqrt{(-5)^2 + (-12)^2}
\][/tex]
[tex]\[
CD = \sqrt{25 + 144}
\][/tex]
[tex]\[
CD = \sqrt{169} = 13 \text{ cm}
\][/tex]
4. Length of side DA:
[tex]\[
D(14, -9.5) \text{ to } A(-4, -2)
\][/tex]
[tex]\[
DA = \sqrt{(14 - (-4))^2 + (-9.5 - (-2))^2}
\][/tex]
[tex]\[
DA = \sqrt{(14 + 4)^2 + (-9.5 + 2)^2}
\][/tex]
[tex]\[
DA = \sqrt{18^2 + (-7.5)^2}
\][/tex]
[tex]\[
DA = \sqrt{324 + 56.25}
\][/tex]
[tex]\[
DA = \sqrt{380.25} = 19.5 \text{ cm}
\][/tex]
#### Step 2: Calculating the perimeter
The perimeter of a rectangle is the sum of all its side lengths:
[tex]\[
\text{Perimeter} = AB + BC + CD + DA
\][/tex]
[tex]\[
\text{Perimeter} = 13 + 19.5 + 13 + 19.5
\][/tex]
[tex]\[
\text{Perimeter} = 65 \text{ cm}
\][/tex]
Thus, the perimeter of rectangle ABCD is [tex]\( \boxed{65} \)[/tex] cm.
