Question

uy || vx. find z.
(diagram with quadrilateral uy and triangle vwx, with uy parallel to vx, segment uy, vx, lengths: u to v is 2z - 11, v to w is z - 2, y to x is 25, x to w is 15)
find z.
z =

Explanation:

Step1: Apply Basic Proportionality Theorem

Since \( UY \parallel VX \), by the Basic Proportionality Theorem (Thales' theorem), the line \( VX \) divides the sides \( WY \) and \( WU \) proportionally. So, \(\frac{WX}{XY}=\frac{WV}{VU}\).
Substituting the given lengths: \(\frac{15}{25}=\frac{z - 2}{2z - 11}\).

Step2: Cross - Multiply to Solve for \( z \)

Cross - multiplying gives \( 15(2z - 11)=25(z - 2) \).
Expanding both sides: \( 30z-165 = 25z - 50 \).
Subtract \( 25z \) from both sides: \( 30z-25z-165=25z - 25z-50 \), which simplifies to \( 5z-165=-50 \).
Add 165 to both sides: \( 5z-165 + 165=-50 + 165 \), so \( 5z=115 \).
Divide both sides by 5: \( z=\frac{115}{5}=23 \).

Answer:

\( z = 23 \)