Question

vw || xz. find xy.
30
10
36
xy =
submit
work it out
not feeling ready yet? these can help:
side lengths and angle measures in similar figures (81)
midsegments of triangles

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{VW} \parallel \overline{XZ} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle YXZ \sim \triangle YWV \) (similar triangles). So the ratios of corresponding sides are equal: \( \frac{YZ}{YV} = \frac{XY}{YW} \).

Step2: Calculate \( YV \) and \( YW \)

\( YV = YZ + ZV = 10 + 30 = 40 \)? Wait, no, wait: Wait, the segment \( YZ = 10 \), \( ZV = 30 \)? Wait, no, looking at the diagram: The left side has \( YZ = 10 \), and the total length from \( Y \) to \( V \) (along the left side) is \( YZ + ZV \)? Wait, no, actually, the sides: \( YZ = 10 \), \( ZV = 30 \)? Wait, no, the vertical segment: the top segment (from \( Y \) to the top) is 30? Wait, no, the diagram shows: \( YZ = 10 \), and the segment from \( Z \) to \( V \) is 30? Wait, no, maybe \( YV = 10 \), \( ZV = 30 \)? Wait, no, let's re-express.

Wait, the triangle: \( Y \) is the bottom left, \( V \) is the top, \( W \) is the bottom right. \( Z \) is on \( YV \), \( X \) is on \( YW \). So \( YZ = 10 \), \( ZV = 30 \), so \( YV = YZ + ZV = 10 + 30 = 40 \)? Wait, no, maybe \( YZ = 10 \), and the length from \( Y \) to \( V \) is \( 10 + 30 = 40 \)? Wait, no, the problem: \( \overline{VW} \parallel \overline{XZ} \), so by similar triangles, \( \frac{YZ}{YV} = \frac{XY}{YW} \). Wait, \( YW = XY + XW = XY + 36 \). \( YZ = 10 \), \( YV = YZ + ZV = 10 + 30 = 40 \)? Wait, no, maybe \( YZ = 10 \), and the segment from \( Y \) to \( V \) is \( 10 \), and from \( Z \) to \( V \) is \( 30 \)? No, that doesn't make sense. Wait, the diagram: the left side has two segments: \( YZ = 10 \), and above \( Z \) to \( V \) is \( 30 \), so total \( YV = 10 + 30 = 40 \). Then \( YW = XY + XW = XY + 36 \). Since \( \triangle YXZ \sim \triangle YWV \), the ratio of sides \( \frac{YZ}{YV} = \frac{XY}{YW} \). So \( \frac{10}{40} = \frac{XY}{XY + 36} \)? Wait, no, that would be \( \frac{10}{40} = \frac{XY}{XY + 36} \), but solving that: \( 10(XY + 36) = 40XY \) → \( 10XY + 360 = 40XY \) → \( 30XY = 360 \) → \( XY = 12 \). Wait, that seems right. Wait, let's check again.

Wait, maybe the similar triangles are \( \triangle YXZ \) and \( \triangle YWV \), with \( XZ \parallel VW \), so corresponding angles are equal, so similarity by AA. So the ratio of \( YZ \) to \( YV \) is equal to the ratio of \( XY \) to \( YW \). \( YZ = 10 \), \( YV = YZ + ZV = 10 + 30 = 40 \). \( YW = XY + XW = XY + 36 \). So \( \frac{10}{40} = \frac{XY}{XY + 36} \). Simplify \( \frac{1}{4} = \frac{XY}{XY + 36} \). Cross-multiplying: \( XY + 36 = 4XY \) → \( 36 = 3XY \) → \( XY = 12 \).

Answer:

\( 12 \)