Question

find the perimeter and area of this figure.
round your answer to the nearest hundredth.

Explanation:

Step1: Label the vertices

Let the vertices 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 \(AB\): \(x_1 = 3,y_1 = 5,x_2 = 5,y_2 = 3\)

$$
LATEXBLOCK0
$$

Since it's a rhombus (all sides equal in a parallelogram with perpendicular diagonals, can be verified by slope - but here symmetry shows equal side lengths), perimeter \(P = 4\times d_{AB}\)

$$ P=4\times\sqrt{8}\approx4\times2.83 = 16.97 $$

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 \(AC\): \(x_1 = 3,y_1 = 5,x_2 = 7,y_2 = 9\)

$$
LATEXBLOCK1
$$

For diagonal \(BD\): \(x_1 = 5,y_1 = 3,x_2 = 9,y_2 = 7\)

$$
LATEXBLOCK2
$$

$$ A=\frac{1}{2}\times\sqrt{32}\times\sqrt{32}=\frac{1}{2}\times32 = 16 $$

Answer:

$P = 16.97$ units, $A = 16.00$ units²