In the standard (x, y) coordinate plane, three vertices of a rhombus are (1, 2), (5, 5), a… — ACT

In the standard $(x, y)$ coordinate plane, three vertices of a rhombus are $(1, 2)$ , $(5, 5)$ , and $(9, 2)$ . If the $y$ -coordinate of the fourth vertex is less than 2, what are the coordinates of the fourth vertex?

$(5, -1)$ Answer
B
$(13, 5)$
C
$(13, -1)$
D
$(6, -2)$
E
$(-3, 5)$

Answer

The coordinates of the fourth vertex are

(5, -1)

The correct answer is the coordinate pair

(5, -1)

. Since opposite sides of a rhombus are parallel and congruent, the vector from

(5, 5)

(1, 2)

(-4, -3)

. Applying this vector to

(9, 2)

gives

(9 - 4, 2 - 3) = (5, -1)

. This point satisfies the requirement that the

y

-coordinate is less than 2, and all four side lengths are exactly 5.

Step-by-Step Solution

Calculate the vector translation between two adjacent vertices of the rhombus.

The vector from vertex

(5, 5)

to vertex

(1, 2)

(1 - 5, 2 - 5) = (-4, -3)

In a rhombus, opposite sides must be parallel and equal in length, meaning the translation from one vertex to another on one side must equal the translation on the opposite side.

Apply the translation vector to the third vertex to find the fourth vertex.

Applying the vector

(-4, -3)

to the vertex

(9, 2)

yields

(9 - 4, 2 - 3) = (5, -1)

This determines the coordinates of the fourth vertex that completes the parallelogram structure.

Verify that the resulting vertex satisfies all conditions of the problem.

The

y

-coordinate of

(5, -1)

-1

, which is less than 2. The side lengths are all equal to 5:

\sqrt{(5-1)^2 + (-1-2)^2} = 5

and

\sqrt{(5-9)^2 + (-1-2)^2} = 5

This confirms that the figure is a rhombus and satisfies the

y

-coordinate constraint.

Key Concept

Using vector translations and distance formulas to determine coordinates of geometric figures on the coordinate plane.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic