In the standard (x, y) coordinate plane, a rhombus ABCD has vertices A(1, 2) and C(7, 10).… — ACT

In the standard $(x, y)$ coordinate plane, a rhombus $ABCD$ has vertices $A(1, 2)$ and $C(7, 10)$ . The length of diagonal $BD$ is half the length of diagonal $AC$ . If the $x$ -coordinate of vertex $B$ is greater than the $x$ -coordinate of vertex $D$ , what is the $y$ -coordinate of vertex $B$ ?

Answer: 4.5

Answer

The

y

-coordinate of vertex

B

4.5

By using the geometric properties of a rhombus, we know that its diagonals bisect each other perpendicularly. The midpoint of diagonal

AC

is calculated as

M(4, 6)

and its length is

10

. Consequently, the perpendicular diagonal

BD

must pass through

M(4, 6)

with a slope of

-\frac{3}{4}

(the negative reciprocal of the slope of

AC

, which is

\frac{4}{3}

). Since the length of

BD

is half the length of

AC

, the length of

BD

5

, meaning vertices

B

and

D

are each a distance of

2.5

units away from

M(4, 6)

. Solving for points along the line

y - 6 = -0.75(x - 4)

at this distance gives

(6, 4.5)

and

(2, 7.5)

. The condition that the

x

-coordinate of

B

is greater than the

x

-coordinate of

D

uniquely determines

B

to be

(6, 4.5)

, yielding a

y

-coordinate of

4.5

Step-by-Step Solution

Calculate the midpoint

M

and the length of diagonal

AC

M = (4, 6)

and

AC = 10

The diagonals of a rhombus bisect each other at their midpoint and their lengths determine the proportions of the shape.

Find the slope and length of diagonal

BD

Slope of

BD

-\frac{3}{4}

, and length is

5

Diagonals of a rhombus are perpendicular, meaning their slopes are negative reciprocals (

m_1 \cdot m_2 = -1

). The problem specifies that the length of

BD

is half of

AC

(

10 \div 2 = 5

Set up equations to find coordinates of

B(x, y)

and

D(x, y)

that are at distance

2.5

from

M(4, 6)

along the line of diagonal

BD

(x - 4)^2 + (y - 6)^2 = 6.25

and

y - 6 = -0.75(x - 4)

Since the diagonals bisect each other, the distance from the midpoint

M

to each of the remaining vertices

B

and

D

is half the length of diagonal

BD

(

5 \div 2 = 2.5

Solve the system of equations for the coordinates.

P_1(6, 4.5)

and

P_2(2, 7.5)

Substituting

y-6

into the distance equation yields

(x-4)^2 + 0.5625(x-4)^2 = 6.25

, which simplifies to

1.5625(x-4)^2 = 6.25 \implies (x-4)^2 = 4 \implies x - 4 = \pm 2

. Thus,

x_1 = 6

(giving

y_1 = 4.5

) and

x_2 = 2

(giving

y_2 = 7.5

Identify vertex

B

using the given coordinate condition.

B = (6, 4.5)

, so the

y

-coordinate is

4.5

The problem states that the

x

-coordinate of

B

is greater than the

x

-coordinate of

D

. Comparing the two solutions, the one with the larger

x

-value (

6 > 2

) must belong to vertex

B

Key Concept

Rhombus Diagonal Properties in the Coordinate Plane

Alternative Method

Alternatively, since the diagonals of a rhombus divide it into four congruent right triangles, we can determine the side length of the rhombus. The legs of these right triangles are half the diagonal lengths:

5

and

2.5

. By the Pythagorean theorem, the square of the side length is

5^2 + 2.5^2 = 31.25

. We can set up distance equations from

B(x, y)

A(1, 2)

and

C(7, 10)

(x-1)^2 + (y-2)^2 = 31.25

and

(x-7)^2 + (y-10)^2 = 31.25

. Subtracting the second equation from the first simplifies to the linear relation

y = -0.75x + 9

, which can then be substituted back into one of the quadratic equations to find

x = 6

x = 2

, yielding

y = 4.5

y = 7.5

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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