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 :

$PQ=\sqrt{(a-0)^2+(b-0)^2}=\sqrt{a^2+b^2},\\\\QR=\sqrt{(a+c-a)^2+(b-b)^2}=\sqrt{c^2}=c,\\\\RS=\sqrt{(c-a-c)^2+(0-b)^2}=\sqrt{a^2+b^2},\\\\SP=\sqrt{(0-c)^2+(0-0)^2}=\sqrt{c^2}=c.$

So, the opposite sides are equal in length.

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

$\textup{slope of PQ, }m_1=\dfrac{b-0}{a-0}=\dfrac{b}{a},\\\\\\\textup{slope of QR, }m_2=\dfrac{b-b}{a+c-a}=0,\\\\\\\textup{slope of RS, }m_3=\dfrac{0-b}{c-a-c}=\dfrac{b}{a},\\\\\\\textup{slope of SP, }m_4=\dfrac{0-0}{0-c}=0.$

So, the slopes of the opposite sides are equal but

$m_1\times m_3=0\neq -1,\\\\m_2\times m_4=0\neq -1.$

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.