find the perimeter and area of this figure p =? units a =? units²

Explanation:

Step1: Assume vertices' coordinates

Let the vertices of the diamond - shaped figure be (A(x_1,y_1)), (B(x_2,y_2)), (C(x_3,y_3)), (D(x_4,y_4)). Suppose (A=(1,6)), (B=(5,3)), (C=(9,6)), (D=(5,9)).

Step2: Use the distance formula (d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}) to find the length of one side

For side (AB): (x_1 = 1,y_1 = 6,x_2 = 5,y_2 = 3). Then (AB=\sqrt{(5 - 1)^2+(3 - 6)^2}=\sqrt{16 + 9}=\sqrt{25}=5). Since the figure is a rhombus (by symmetry), all sides are equal.

Step3: Calculate the perimeter (P)

The perimeter of a rhombus (P = 4s), where (s) is the side - length. So (P=4\times5 = 20) units.

Step4: Calculate the area (A)

The area of a rhombus (A=\frac{1}{2}d_1d_2), where (d_1) and (d_2) are the lengths of the diagonals. The length of diagonal (AC): (A=(1,6)), (C=(9,6)), so (d_1=\vert9 - 1\vert = 8). The length of diagonal (BD): (B=(5,3)), (D=(5,9)), so (d_2=\vert9 - 3\vert = 6). Then (A=\frac{1}{2}\times8\times6=24) square units.

Answer:

(P = 20) units, (A = 24) units²