Question
Given that ABCD is a rectangle with AO = 4x - 3 and BO = 5x - 10, what is the value of x?
a. x = 17/6
b. x = -13
c. x = 7
d. x = -4/3
a. x = 17/6
b. x = -13
c. x = 7
d. x = -4/3
Answers
Answered by
GPT-4o mini
In a rectangle, the diagonals are equal in length. Given points A and B are opposite corners of the rectangle, we can set the lengths of the diagonals equal to each other.
Since \( AO \) and \( BO \) are the lengths of the diagonals from points A to point O and from point B to point O, we can write the equation:
\[
AO = BO
\]
Substituting the expressions given for \( AO \) and \( BO \):
\[
4x - 3 = 5x - 10
\]
Now, let's solve for \( x \):
1. Rearrange the equation to get all terms involving \( x \) on one side and constant terms on the other side:
\[
4x - 5x = -10 + 3
\]
2. Simplify both sides:
\[
-x = -7
\]
3. Multiply both sides by -1:
\[
x = 7
\]
Thus, the value of \( x \) is \( \boxed{7} \), which corresponds to option c.
Since \( AO \) and \( BO \) are the lengths of the diagonals from points A to point O and from point B to point O, we can write the equation:
\[
AO = BO
\]
Substituting the expressions given for \( AO \) and \( BO \):
\[
4x - 3 = 5x - 10
\]
Now, let's solve for \( x \):
1. Rearrange the equation to get all terms involving \( x \) on one side and constant terms on the other side:
\[
4x - 5x = -10 + 3
\]
2. Simplify both sides:
\[
-x = -7
\]
3. Multiply both sides by -1:
\[
x = 7
\]
Thus, the value of \( x \) is \( \boxed{7} \), which corresponds to option c.