Question

6 fill in the blank 6 points solve for x. (5x - 25)° (3x + 9)° type your answer.. 7 fill in the blank 6 points quadrilateral wxyz is a rectangle. if zx = 2x + 1 and zp = 11, find x. z y p w x type your answer..

Explanation:

Question 6

Step1: Identify angle relationship

The two angles \((5x - 25)^\circ\) and \((3x + 9)^\circ\) are equal (corresponding angles or alternate interior angles, as the lines are parallel and cut by a transversal, so they are equal). So set \(5x - 25 = 3x + 9\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(5x - 3x - 25 = 3x - 3x + 9\) → \(2x - 25 = 9\).
Add 25 to both sides: \(2x - 25 + 25 = 9 + 25\) → \(2x = 34\).
Divide by 2: \(\frac{2x}{2} = \frac{34}{2}\) → \(x = 17\).

Step1: Recall rectangle diagonal property

In a rectangle, diagonals are equal and bisect each other. So \(ZX = 2 \times ZP\) (since \(P\) is the midpoint of diagonal \(ZX\)).

Step2: Substitute values and solve

Given \(ZX = 2x + 1\) and \(ZP = 11\), so \(2x + 1 = 2 \times 11\) → \(2x + 1 = 22\).
Subtract 1: \(2x + 1 - 1 = 22 - 1\) → \(2x = 21\).
Divide by 2: \(\frac{2x}{2} = \frac{21}{2}\) → Wait, correction: Wait, in a rectangle, diagonals bisect each other, so \(ZX = 2 \times ZP\)? Wait, no: \(ZX\) is a diagonal, and \(ZP\) is half of the diagonal (since diagonals bisect each other). So \(ZX = 2 \times ZP\)? Wait, no: \(ZP\) is a segment from vertex \(Z\) to the midpoint \(P\) of the diagonal \(WX\) (wait, no, in rectangle \(WXYZ\), diagonals \(WY\) and \(ZX\) intersect at \(P\), so \(ZP = PX\) and \(ZX = ZP + PX = 2 \times ZP\). So \(ZX = 2 \times ZP\). So \(2x + 1 = 2 \times 11\) → \(2x + 1 = 22\) → \(2x = 21\) → \(x = 10.5\)? Wait, no, wait: Wait, the problem says \(ZX = 2x + 1\) and \(ZP = 11\). Wait, maybe I mixed up the diagonals. Wait, in rectangle \(WXYZ\), diagonals \(WX\) and \(ZY\)? No, vertices are \(W, X, Y, Z\), so diagonals are \(WY\) and \(ZX\). So they intersect at \(P\), so \(ZP = \frac{1}{2}ZX\). Therefore, \(ZX = 2 \times ZP\). So \(2x + 1 = 2 \times 11\) → \(2x + 1 = 22\) → \(2x = 21\) → \(x = 10.5\). Wait, but maybe the problem has a typo, or I misread. Wait, the problem says \(ZX = 2x + 1\) and \(ZP = 11\). So if \(P\) is the midpoint, then \(ZX = 2 \times ZP\), so \(2x + 1 = 22\) → \(2x = 21\) → \(x = 10.5\) (or \(\frac{21}{2}\)).
Wait, maybe I made a mistake. Let's recheck: In a rectangle, diagonals are equal and bisect each other. So \(ZP = \frac{1}{2}ZX\), so \(ZX = 2 \times ZP\). So \(2x + 1 = 2 \times 11\) → \(2x + 1 = 22\) → \(2x = 21\) → \(x = 10.5\).

Answer:

\(17\)

Question 7