Question

find the perimeter and area of this figure.
p = ? units
a = units²
round your answer to the nearest hundredth.

Explanation:

Step1: Label the points

Let the points be \(A(3,5)\), \(B(5,3)\), \(C(7,9)\), \(D(9,7)\).

Step2: Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(AB\): \(x_1 = 3,y_1 = 5,x_2 = 5,y_2 = 3\)
\(AB=\sqrt{(5 - 3)^2+(3 - 5)^2}=\sqrt{4 + 4}=\sqrt{8}\approx2.83\)
Since it's a rhombus (all sides equal), perimeter \(P = 4\times AB\)
\(P=4\times\sqrt{8}\approx4\times2.83 = 11.32\)

Step3: Use the formula for the area of a rhombus \(A=\frac{1}{2}d_1d_2\)

Find the lengths of the diagonals.
For diagonal \(AC\): \(x_1 = 3,y_1 = 5,x_2 = 7,y_2 = 9\)
\(AC=\sqrt{(7 - 3)^2+(9 - 5)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.66\)
For diagonal \(BD\): \(x_1 = 5,y_1 = 3,x_2 = 9,y_2 = 7\)
\(BD=\sqrt{(9 - 5)^2+(7 - 3)^2}=\sqrt{16+16}=\sqrt{32}\approx5.66\)
\(A=\frac{1}{2}\times\sqrt{32}\times\sqrt{32}=\frac{1}{2}\times32 = 16\)

Answer:

\(P = 11.32\) units, \(A=16\) units²