In the standard (x,y) coordinate plane, a square has two adjacent vertices at A(1, 4) and … — ACT

In the standard $(x,y)$ coordinate plane, a square has two adjacent vertices at $A(1, 4)$ and $B(4, 8)$ . If the square lies entirely in the first quadrant, what is the $x$ -coordinate of the vertex diagonally opposite to $A$ ?

Answer: 8

Answer

The

x

-coordinate of the vertex diagonally opposite to

A

8

The vector from

A(1,4)

B(4,8)

\vec{AB} = (3,4)

with length

5

. Since the adjacent sides of a square must be perpendicular and equal in length, the vector translation must be either

(4, -3)

(-4, 3)

. Applying the translation

(4, -3)

B

and

A

gives

C(8, 5)

and

D(5, 1)

, both of which lie strictly in the first quadrant. The diagonally opposite vertex to

A

C

, and its

x

-coordinate is

8

Step-by-Step Solution

Calculate the vector representing side

AB

and its length.

\vec{AB} = (3, 4)

, which has a length of

\sqrt{3^2 + 4^2} = 5

This determines the length and orientation of the square's base.

Determine the perpendicular vectors of equal length to find the directions of the adjacent sides.

The perpendicular vectors of length

5

are

(4, -3)

and

(-4, 3)

The adjacent sides of a square are perpendicular and equal in length to side

AB

Find the coordinates of the other two vertices for both possible positions of the square.

Position 1 vertices:

C = B + (4, -3) = (8, 5)

and

D = A + (4, -3) = (5, 1)

. Position 2 vertices:

C' = B + (-4, 3) = (0, 11)

and

D' = A + (-4, 3) = (-3, 7)

Adding the perpendicular vectors to the coordinates of the base vertices yields the remaining vertices.

Determine which square lies entirely in the first quadrant and identify the target coordinate.

The square

ABCD

with vertices

C(8, 5)

and

D(5, 1)

lies entirely in the first quadrant because all

x

- and

y

-coordinates are positive. The vertex diagonally opposite to

A

C(8, 5)

, which has an

x

-coordinate of

8

The alternative square

ABC'D'

has a vertex at

D'(-3, 7)

, which lies in the second quadrant.

Key Concept

Using vector methods to determine the coordinates of a geometric figure on the coordinate plane under specific quadrant constraints.

Estimated Time:2m 0s

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