Question

1. find the perimeter of quadrilateral abcd with vertices a(-2, -2), b(-1, -3), c(5, 3), d(4, -2).

Explanation:

Step1: Use distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate length of AB

For points $A(-2,-2)$ and $B(-1,-3)$, $d_{AB}=\sqrt{(-1+2)^2+(-3 + 2)^2}=\sqrt{1^2+(-1)^2}=\sqrt{1 + 1}=\sqrt{2}$.

Step3: Calculate length of BC

For points $B(-1,-3)$ and $C(5,3)$, $d_{BC}=\sqrt{(5 + 1)^2+(3 + 3)^2}=\sqrt{6^2+6^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}$.

Step4: Calculate length of CD

For points $C(5,3)$ and $D(4,-2)$, $d_{CD}=\sqrt{(4 - 5)^2+(-2 - 3)^2}=\sqrt{(-1)^2+(-5)^2}=\sqrt{1 + 25}=\sqrt{26}$.

Step5: Calculate length of DA

For points $D(4,-2)$ and $A(-2,-2)$, $d_{DA}=\sqrt{(-2 - 4)^2+(-2+2)^2}=\sqrt{(-6)^2+0^2}=6$.

Step6: Calculate perimeter

$P=d_{AB}+d_{BC}+d_{CD}+d_{DA}=\sqrt{2}+6\sqrt{2}+\sqrt{26}+6=7\sqrt{2}+\sqrt{26}+6$.

Answer:

$7\sqrt{2}+\sqrt{26}+6$