Question

The coordinates of the vertices for the figure HIJK are H(0, 5), I(3, 3), J(4, –1), and K(1, 1). To determine if it is a parallelogram, use the converse of the parallelogram diagonal theorem. This states that if the diagonals , then the quadrilateral is a parallelogram. The midpoint of HJ is and the midpoint of IK is (2, 2). Therefore, HIJK is a parallelogram because the diagonals , which means they bisect each other.

Answer 1

   →Coordinates of the vertices for the figure HIJK are H(0, 5), I(3, 3), J(4, –1), and K(1, 1).

Now the converse of Parallelogram diagonal theorem states that if opposite sides of Quadrilateral  are congruent, then it is a parallelogram.

it is given that diagonals of the quadrilateral HIJK meet each other at a common point of intersection i.e at (2,2).

NOW USING THE DIS TANCE FORMULA BETWEEN TWO POINTS  i.e between $(x_{1},y_{1}) and (x_{2},y_{2})=\sqrt{({x_2-x_1})^2+({y_2-y_1})^2}$

So by applying the same formula we get

HI=${\sqrt{(3-0)^2+(3-5)^2)}$=√9+4=√13

IJ=${\sqrt{(4-3)^2+(-1-3)^2}$=√1+16=√17

JK=$\sqrt{(4-1)^2+(-1-1)^2}$=√9+4=√13

KI=$\sqrt{(1-0)^2+(1-5)^2}$=√1+16=√17

HJ=$\sqrt{(4-0)^2+(-1-5)^2}$=√16+36=√52=2√13

KI=$\sqrt{(3-1)^2+(3-1)^2}$=√4+4=√8

HI=√13,  IJ=√17, JK=√17,HI=√13

Now consider opposite sides of a Quadrilateral

HI=JK=√13

HK=JI=√17

So we have seen in the figure that opposite sides of the quadrilateral are congruent .It means it is a Parallelogram.

2nd Method.

In ΔHKJ and ΔHIJ

HK=IJ=√17

HI=KJ=√13

HJ=HJ [Common}

ΔHKJ ≅ ΔHIJ [SSS]

Now consider ΔHKI AND ΔKJI

HK=IJ=√17

HI=KJ=√13

KI=KI [Common]

ΔHKI ≅ ΔKJI [SSS]

Now the converse of Parallelogram diagonal theorem also states that if diagonals of a Quadrilateral divides it into two congruent triangles then it is a parallelogram.

Hence the given Quadrilateral is a Parallelogram.

Answer 2

The answer is

bisects each other

(2,2)

have the same mid point

ya welcome =)