Question

select the correct answer. what is the perimeter of the polygon in the diagram? a. 2\sqrt{(a - b)^2} b. 4(a + b)^2 c. 2(a^2 + b^2) d. 4\sqrt{a^2 + b^2}

Explanation:

Step1: Use distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Consider the points $(0,0)$ and $(a,0)$. The distance between $(0,0)$ and $(a,0)$ is $d_1=\sqrt{(a - 0)^2+(0 - 0)^2}=a$. Consider the points $(0,0)$ and $(0,b)$. The distance between two adjacent non - collinear vertices, say $(0,0)$ and $(a,b)$ (using the distance formula) is $d=\sqrt{(a - 0)^2+(b - 0)^2}=\sqrt{a^{2}+b^{2}}$.

Step2: Analyze the symmetry

The polygon is symmetric about both the x - axis and y - axis. All the side - lengths of the polygon are equal. The length of one side, for example, the side connecting $(0,0)$ and $(a,0)$ and $(0,b)$ can be found using the distance formula between $(0,0)$ and $(a,b)$ which is $\sqrt{(a - 0)^2+(b - 0)^2}=\sqrt{a^{2}+b^{2}}$. Since the polygon has 4 equal sides.

Step3: Calculate the perimeter

The perimeter $P$ of a polygon with 4 equal sides of length $s=\sqrt{a^{2}+b^{2}}$ is $P = 4s=4\sqrt{a^{2}+b^{2}}$.

Answer:

D. $4\sqrt{a^{2}+b^{2}}$