Rectangle upper A upper B upper C upper D is shown with point upper O in the center. Diagonal segments upper A upper C and upper B upper D intersect at point upper O, in the center of the rectangle. Some line segments have measurements.

Question

The rectangle ABCD is given as shown. Find x if CO=−3x and BD=x+21 .

(1 point)
Responses

x=3
x equals 3

x=−214
x equals negative Start Fraction 21 over 4 End Fraction

x=−425
x equals negative Start Fraction 42 over 5 End Fraction

x=−3

Answer 1

In rectangle ABCD, the diagonals AC and BD intersect at point O, which is the center of the rectangle. The lengths of the diagonals in a rectangle are equal, so we have:

1. \( AC = BD \)

Given the information:

• \( CO = -3x \)
• \( BD = x + 21 \)

Since point O is the midpoint of both diagonals, we can express the entire diagonal lengths in terms of CO and BD.

The entire diagonal AC can be expressed as: \[ AC = 2 \times CO = 2 \times (-3x) = -6x \]

For diagonal BD: \[ BD = x + 21 \]

Setting these equal to each other: \[ -6x = x + 21 \]

Now, we can solve for \( x \):

1. Add \( 6x \) to both sides: \[ 0 = 7x + 21 \]

2. Subtract 21 from both sides: \[ -21 = 7x \]

3. Divide by 7: \[ x = -3 \]

Thus, the value of \( x \) is: x = -3

Therefore, the correct response is \( x = -3 \).