If XZ = 7x + 1 and PZ = 4x - 1, what is XP?

1. Identify the shape and properties: The image shows a quadrilateral WXYZ with its diagonals XZ and WY intersecting at point P. The arrows on the sides indicate that WX is parallel to YZ and WY is parallel to XZ. This means that WXYZ is a parallelogram. A key property of parallelograms is that their diagonals bisect each other. This means that the point of intersection P divides each diagonal into two equal segments.

2. Set up equations based on diagonal properties: Since the diagonals bisect each other, we have:
$XP = PZ$
$WP = PY$

3. Use the given information: We are given the lengths of segments of the diagonal XZ in terms of x:
$XZ = 7x + 1$
$PZ = 4x - 1$

Since P is the midpoint of XZ, we know that $XP = PZ$. Therefore, we can set the expression for PZ equal to itself, but this doesn't help us find x. Instead, we know that the entire diagonal XZ is made up of XP and PZ. So, $XZ = XP + PZ$. Since $XP = PZ$, we can write $XZ = 2 PZ$ or $XZ = 2 XP$. Let's use $XZ = 2 * PZ$ because we have an expression for PZ.

$$7x + 1 = 2(4x - 1)$$

4. Solve for x: Now, we solve the equation for x:
$$7x + 1 = 8x - 2$$
Subtract $7x$ from both sides:
$$1 = x - 2$$
Add 2 to both sides:
$$3 = x$$
So, $x = 3$.

5. Calculate the length of XP: We need to find the length of XP. We know that $XP = PZ$. We can use the expression for PZ and substitute the value of x we found:
$$PZ = 4x - 1$$
$$PZ = 4(3) - 1$$
$$PZ = 12 - 1$$
$$PZ = 11$$

Since $XP = PZ$, then $XP = 11$.

Alternatively, we can find XP using the expression for XZ and the fact that $XP = XZ / 2$:
$$XZ = 7x + 1$$
$$XZ = 7(3) + 1$$
$$XZ = 21 + 1$$
$$XZ = 22$$
$$XP = XZ / 2$$
$$XP = 22 / 2$$
$$XP = 11$$

Both methods yield the same result.