Question

1. parallelogram klmn k(-1,1) l(4,1) m(2,-2) n(-3,-2)

Explanation:

Step1: Calculate the length of \(KL\)

Use the distance formula for points with the same \(y\) - coordinate: \(d=\vert x_2 - x_1\vert\).
For \(K(-1,1)\) and \(L(4,1)\), \(KL=\vert4-(-1)\vert = 5\).

Step2: Calculate the length of \(LM\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(L(4,1)\) and \(M(2,-2)\), \(LM=\sqrt{(2 - 4)^2+(-2 - 1)^2}=\sqrt{(-2)^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\).

Step3: Calculate the perimeter

The perimeter \(P\) of a parallelogram is \(P = 2(a + b)\), where \(a\) and \(b\) are the side lengths.
Here \(a = 5\) and \(b=\sqrt{13}\), so \(P=2(5+\sqrt{13})=10 + 2\sqrt{13}\).

Step4: Calculate the area

The base \(b = KL=5\). The height \(h\) (vertical distance between \(y = 1\) and \(y=-2\)) is \(h=\vert1-(-2)\vert = 3\).
The area \(A\) of a parallelogram is \(A=\text{base}\times\text{height}\), so \(A=5\times3 = 15\).

Answer:

Perimeter: \(10 + 2\sqrt{13}\); Area: \(15\)