Question

Use the image to answer the question.

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

x=7
x equals 7

x=−13
x equals negative 13

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

x=176
x equals Start Fraction 17 over 6 End Fraction
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page 15 of 15

Answer 1

In a rectangle, the diagonals bisect each other at their midpoint. Therefore, the lengths of the segments from the vertices to the intersection point (O) of the diagonals must be equal.

In this case, we have the lengths:

- \( AO = 4x - 3 \)
- \( BO = 5x - 10 \)

Since \( AO \) and \( BO \) are equal, we can set up the following equation:

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

To solve for \( x \), first, rearrange the equation:

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

\[
-3 = x - 10
\]

2. Next, add 10 to both sides:

\[
7 = x
\]

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

Thus, the answer is:

\( x = 7 \)