Question

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 the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For example, between \((a,0)\) and \((0,b)\), \(d=\sqrt{(a - 0)^2+(0 - b)^2}=\sqrt{a^{2}+b^{2}}\).

Step2: Calculate the perimeter

Since the polygon is a rhombus (all sides are equal). The perimeter \(P = 4\times\) side - length. The side - length is \(\sqrt{a^{2}+b^{2}}\), so \(P = 4\sqrt{a^{2}+b^{2}}\).

Answer:

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