Question

given that abcd is a rectangle with ao = 4x - 3 and bo = 5x - 10, what is the value of x? (1 point) \\( \bigcirc \\ x = 7 \\) \\( \bigcirc \\ x = \frac{17}{6} \\) \\( \bigcirc \\ x = -\frac{4}{3} \\) \\( \bigcirc \\ x = -13 \\)

Explanation:

Step1: Recall rectangle diagonal property

In a rectangle, the diagonals are equal and bisect each other, so \(AO = BO\).

Step2: Set up the equation

Set \(4x - 3 = 5x - 10\).

Step3: Solve for x

Subtract \(4x\) from both sides: \(-3 = x - 10\).
Add 10 to both sides: \(x = -3 + 10 = 7\)? Wait, no, wait. Wait, let's do it again. Wait, \(4x - 3 = 5x - 10\). Subtract \(4x\) from both sides: \(-3 = x - 10\). Then add 10 to both sides: \(x = 7\)? Wait, but let's check. Wait, maybe I made a mistake. Wait, \(4x - 3 = 5x - 10\). Let's move terms: \( -3 + 10 = 5x - 4x\), so \(7 = x\). Wait, but let's check the options. Wait, the first option is \(x = 7\). Wait, but let's verify. If \(x = 7\), then \(AO = 4*7 - 3 = 28 - 3 = 25\), \(BO = 5*7 - 10 = 35 - 10 = 25\). So they are equal. So that works. Wait, but maybe I misread the problem. Wait, the problem says \(AO = 4x - 3\) and \(BO = 5x - 10\). In a rectangle, diagonals bisect each other, so \(AO = BO\). So equation is \(4x - 3 = 5x - 10\). Solving: \( -3 + 10 = 5x - 4x\) => \(7 = x\). So \(x = 7\).

Wait, but let me check again. \(4x - 3 = 5x - 10\). Subtract \(4x\) from both sides: \(-3 = x - 10\). Add 10 to both sides: \(x = 7\). Yes, that's correct. So the answer is \(x = 7\).

Answer:

\(x = 7\) (corresponding to the option "x = 7")