Question

1. quadrilateral jklm has vertices j(2, 4), k(6, 1), l(2, - 2) and m(- 2, 1). what type of quadrilateral is jklm? (5 marks) explain more than one way to verify the property of a quadrilateral

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $JK$ with $J(2,4)$ and $K(6,1)$:
$d_{JK}=\sqrt{(6 - 2)^2+(1 - 4)^2}=\sqrt{4^2+( - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$
For side $KL$ with $K(6,1)$ and $L(2,-2)$:
$d_{KL}=\sqrt{(2 - 6)^2+( - 2 - 1)^2}=\sqrt{( - 4)^2+( - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$
For side $LM$ with $L(2,-2)$ and $M(-2,1)$:
$d_{LM}=\sqrt{( - 2 - 2)^2+(1+2)^2}=\sqrt{( - 4)^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5$
For side $MJ$ with $M(-2,1)$ and $J(2,4)$:
$d_{MJ}=\sqrt{(2 + 2)^2+(4 - 1)^2}=\sqrt{4^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5$
All four - sides are equal.

Step2: Calculate the slopes of the diagonals

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For diagonal $JL$ with $J(2,4)$ and $L(2,-2)$:
The slope of $JL$ is $m_{JL}=\frac{-2 - 4}{2 - 2}$, which is undefined (vertical line).
For diagonal $KM$ with $K(6,1)$ and $M(-2,1)$:
The slope of $KM$ is $m_{KM}=\frac{1 - 1}{-2 - 6}=0$ (horizontal line).
Since the diagonals are perpendicular (one has a slope of 0 and the other is undefined), and all four sides are equal.

Answer:

Quadrilateral $JKLM$ is a rhombus.