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A rectangle upper A upper B upper C upper D has diagonals between upper A upper C and upper B upper D. The diagonals insect at upper O, in the center. Segment upper A upper O is labeled 4 x minus 3. Segment upper B upper O is labeled 5 x minus 10.

Given that ABCD
is a rectangle with AO=4x−3
and BO=5x−10
, what is the value of x?

(1 point)
Responses which answer is correct and why

x=−43
x equals negative Start Fraction 4 over 3 End Fraction

x=176
x equals Start Fraction 17 over 6 End Fraction

x=7
x equals 7

x=−13
x equals negative 13
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1 answer

In a rectangle, the diagonals bisect each other. This means that segments AO and BO must be equal in length at point O. So, we set up the equation based on the given segments:

\[ AO = BO \]

Substituting the expressions for AO and BO, we get:

\[ 4x - 3 = 5x - 10 \]

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

  1. Subtract \( 4x \) from both sides:

\[ -3 = x - 10 \]

  1. Add 10 to both sides:

\[ 7 = x \]

Thus, the value of \( x \) is \( 7 \).

The correct answer is x = 7.

This solution works because it is based on the property of rectangles that states the diagonals bisect each other, leading to the equality of the segments.