Question

If the coordinates of a quadrilateral composed in the first quadrant of the coordinate plane are (0,0) (a, b), (a + c, b), and (c, o), which is the most accurate classification of the quadrilateral? A. Trapezoid B. Rectangle C. Parallelogram D. Square

Answer 1

Answer: The correct option is (C) Parallelogram.

Step-by-step explanation: Given that the co-ordinates of the vertices of a quadrilateral composed in the first quadrant of the coordinate plane are (0,0) (a, b), (a + c, b), and (c, 0).

We are to select the most accurate classification of the quadrilateral.

Let the co-ordinates of the vertices of the quadrilateral are P(0,0), Q(a, b), R(a + c, b), and S(c, 0).

The lengths of the sides are calculated using distance formula as follows :

`PQ=√((a-0)^2+(b-0)^2)=√(a^2+b^2),\\\\QR=√((a+c-a)^2+(b-b)^2)=√(c^2)=c,\\\\RS=√((c-a-c)^2+(0-b)^2)=√(a^2+b^2),\\\\SP=√((0-c)^2+(0-0)^2)=√(c^2)=c.`

So, the opposite sides are equal in length.

And, the slopes of the sides are calculated as follows :

`\textup{slope of PQ, }m_1=(b-0)/(a-0)=(b)/(a),\\\\\\\textup{slope of QR, }m_2=(b-b)/(a+c-a)=0,\\\\\\\textup{slope of RS, }m_3=(0-b)/(c-a-c)=(b)/(a),\\\\\\\textup{slope of SP, }m_4=(0-0)/(0-c)=0.`

So, the slopes of the opposite sides are equal but

`m_1* m_3=0\\eq -1,\\\\m_2* m_4=0\\eq -1.`

Thus, the opposite sides of the quadrilateral PQRS are equal and parallel and so PQRS is a PARALLELOGRAM.

Option (C) is CORRECT.

Answer 2

The figure or the polygon that is portrayed by the following coordinates of (0,0) (a, b), (a + c, b), and (c, o), is called a parallelogram. By definition, a parallelogram primarily consists of having sides that are considered as parallel and at least a pair of sides and angles inside is congruent.