Question

20va v, geometry sem a- a
sing triangle similarity theorems
if $overleftrightarrow{cd} \parallel \overleftrightarrow{xz}$ and $cx = 5$ units, what is $dz$?
2 units
5 units
3 units
4 units

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{CD} \parallel \overline{XZ} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle YCD \sim \triangle YXZ \). This gives the proportion \( \frac{YC}{YX}=\frac{YD}{YZ} \).

Step2: Calculate \( YC \) and \( YX \)

Given \( YC = 25 - CX \)? Wait, no, \( YC \) is 25? Wait, looking at the diagram: \( YC \) is from Y to C, length 25? Wait, no, \( YX \) is \( YC + CX \), \( YC = 25 \)? Wait, no, the sides: \( YC \) is 25? Wait, \( YD = 20 \), \( CD = 15 \), \( XZ = 18 \). Wait, actually, the correct proportion is \( \frac{YC}{YX}=\frac{CD}{XZ}=\frac{YD}{YZ} \). Wait, \( YC = 25 - CX \)? No, \( YX = YC + CX \), \( YC = 25 \)? Wait, the diagram: \( YC \) is 25? Wait, \( CX \) is 5, so \( YX = YC + CX = 25 + 5 = 30 \)? Wait, no, maybe \( YC = 25 \), \( CX = 5 \), so \( YX = YC + CX = 30 \). Then \( CD = 15 \), \( XZ = 18 \). Wait, no, the ratio of similarity: \( \frac{CD}{XZ}=\frac{15}{18}=\frac{5}{6} \). Then \( \frac{YD}{YZ}=\frac{5}{6} \). \( YD = 20 \), so \( \frac{20}{YZ}=\frac{5}{6} \), so \( YZ = 24 \). Then \( DZ = YZ - YD = 24 - 20 = 4 \). Wait, let's check again.

Wait, the correct approach: Since \( CD \parallel XZ \), \( \triangle YCD \sim \triangle YXZ \) by AA similarity (corresponding angles equal because of parallel lines). So the ratio of corresponding sides is equal. So \( \frac{YC}{YX}=\frac{CD}{XZ}=\frac{YD}{YZ} \). \( YC = 25 \)? Wait, no, \( CX = 5 \), so \( YX = YC + CX \). Wait, maybe \( YC = 25 \), \( CX = 5 \), so \( YX = 25 + 5 = 30 \). \( CD = 15 \), \( XZ = 18 \). Then \( \frac{YC}{YX}=\frac{25}{30}=\frac{5}{6} \), \( \frac{CD}{XZ}=\frac{15}{18}=\frac{5}{6} \), so that's consistent. Then \( \frac{YD}{YZ}=\frac{5}{6} \). \( YD = 20 \), so \( \frac{20}{YZ}=\frac{5}{6} \), so \( YZ = 24 \). Then \( DZ = YZ - YD = 24 - 20 = 4 \). So DZ is 4 units.

Answer:

4 units