Question

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