Question

part a identify the figure with the vertices k(-1,1), l(3,4), m(6,0), and n(2,-3). select choice part b find the perimeter and area of the figure. perimeter: units area: units²

Explanation:

Step1: Calculate the lengths of the sides using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$

For side $KL$ where $K(-1,1)$ and $L(3,4)$:
$d_{KL}=\sqrt{(3 - (-1))^2+(4 - 1)^2}=\sqrt{4^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5$
For side $LM$ where $L(3,4)$ and $M(6,0)$:
$d_{LM}=\sqrt{(6 - 3)^2+(0 - 4)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$
For side $MN$ where $M(6,0)$ and $N(2,-3)$:
$d_{MN}=\sqrt{(2 - 6)^2+(-3 - 0)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16+9}=\sqrt{25}=5$
For side $NK$ where $N(2,-3)$ and $K(-1,1)$:
$d_{NK}=\sqrt{(-1 - 2)^2+(1 - (-3))^2}=\sqrt{(-3)^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$

Step2: Determine the figure type

Since all four - side lengths are equal ($d_{KL}=d_{LM}=d_{MN}=d_{NK} = 5$), the figure is a rhombus.

Step3: Calculate the perimeter

The perimeter $P$ of a polygon is the sum of the lengths of its sides. For a rhombus with side length $s$, $P = 4s$. Here $s = 5$, so $P=4\times5 = 20$ units.

Step4: Calculate the area

The area $A$ of a rhombus can be found using the formula $A=\frac{1}{2}\times d_1\times d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.
The mid - point formula is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. The diagonals of the rhombus are $KM$ and $LN$.
The mid - point of $KM$: $(\frac{-1+6}{2},\frac{1 + 0}{2})=(\frac{5}{2},\frac{1}{2})$
The mid - point of $LN$: $(\frac{3+2}{2},\frac{4-3}{2})=(\frac{5}{2},\frac{1}{2})$, so the diagonals bisect each other.
The length of diagonal $KM$ where $K(-1,1)$ and $M(6,0)$:
$d_{KM}=\sqrt{(6 - (-1))^2+(0 - 1)^2}=\sqrt{7^2+(-1)^2}=\sqrt{49 + 1}=\sqrt{50}=5\sqrt{2}$
The length of diagonal $LN$ where $L(3,4)$ and $N(2,-3)$:
$d_{LN}=\sqrt{(2 - 3)^2+(-3 - 4)^2}=\sqrt{(-1)^2+(-7)^2}=\sqrt{1 + 49}=\sqrt{50}=5\sqrt{2}$
$A=\frac{1}{2}\times d_{KM}\times d_{LN}=\frac{1}{2}\times5\sqrt{2}\times5\sqrt{2}=\frac{1}{2}\times50 = 25$ square units.

Answer:

Part A: Rhombus
Part B:
perimeter: 20 units
area: 25 units²