Three vertices of a parallelogram are A(1, 3), B(-2, -1), and C(4, -1) in the standard (x,… — ACT

Three vertices of a parallelogram are $A(1, 3)$ , $B(-2, -1)$ , and $C(4, -1)$ in the standard $(x, y)$ coordinate plane. If the fourth vertex, $D$ , is located in the fourth quadrant, what is the $y$ -coordinate of $D$ ?

Cevap: -5

Cevap

The

y

-coordinate of the fourth vertex of the parallelogram in the fourth quadrant is

-5

By applying the midpoint formula to the diagonals of the three possible parallelograms formed by the vertices

A(1, 3)

B(-2, -1)

, and

C(4, -1)

, we find the candidate points for the fourth vertex

D

to be

(7, 3)

(-5, 3)

, and

(1, -5)

. The point

(1, -5)

is the only candidate that lies in the fourth quadrant, where

x > 0

and

y < 0

. Therefore, the

y

-coordinate of the fourth vertex is

-5

Adım Adım Çözüm

Set up equations based on the property that the diagonals of a parallelogram bisect each other (have the same midpoint).

Three possible configurations of diagonals lead to three potential sets of coordinates for the fourth vertex

D(x, y)

(7, 3)

(-5, 3)

, and

(1, -5)

Three points in a coordinate plane can form three distinct parallelograms depending on which pairs are connected as diagonals.

Identify the signs of the coordinates for each potential vertex to determine which quadrant it lies in.

The point

(7, 3)

is in Quadrant I (

x > 0, y > 0

). The point

(-5, 3)

is in Quadrant II (

x < 0, y > 0

). The point

(1, -5)

is in Quadrant IV (

x > 0, y < 0

The fourth quadrant is defined by positive

x

-values and negative

y

-values.

Select the correct coordinate value corresponding to the question's requirement.

The

y

-coordinate of the vertex in the fourth quadrant,

(1, -5)

, is

-5

The question asks specifically for the

y

-coordinate of the fourth vertex

D

Anahtar Kavram

Finding a missing vertex of a parallelogram on the coordinate plane using midpoint properties

Soru

Cevap

Adım Adım Çözüm

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