Question

using triangle similarity theorems
what is the length of \\(\overline{dc}\\)?
2 units
3 units
9 units
6 units

Explanation:

Step1: Identify Midline Theorem

Since \( DE \parallel BC \) (marked by congruent segments), \( DE \) is a midline? Wait, no, check the ratios. \( AE = 12 \), \( EB = 4 \), so \( AB = AE + EB = 16 \)? Wait, no, \( AE = 12 \), \( EB = 4 \), so \( \frac{AE}{AB}=\frac{12}{12 + 4}=\frac{12}{16}=\frac{3}{4} \)? Wait, no, maybe \( D \) and \( E \) are midpoints? Wait, no, the segments on \( AC \): \( AD = 9 \), let \( DC = x \), so \( AC = 9 + x \). By the Basic Proportionality Theorem (Thales' theorem), if \( DE \parallel BC \), then \( \frac{AE}{EB}=\frac{AD}{DC} \)? Wait, no, Thales' theorem is \( \frac{AE}{AB}=\frac{AD}{AC} \) if \( DE \parallel BC \). Wait, let's re - check the diagram. The marks on \( DE \) and \( BC \) suggest \( DE \parallel BC \), so triangles \( ADE \) and \( ABC \) are similar? Wait, no, the ratio of \( AE \) to \( AB \): \( AE = 12 \), \( EB = 4 \), so \( AB=12 + 4 = 16 \), \( AE/AB = 12/16 = 3/4 \). Then \( AD/AC=3/4 \), \( AD = 9 \), so \( 9/AC = 3/4 \), \( AC = 12 \), so \( DC=AC - AD=12 - 9 = 3 \)? Wait, no, maybe I mixed up the segments. Wait, maybe \( E \) divides \( AB \) in the ratio \( AE:EB = 12:4=3:1 \), and \( D \) divides \( AC \) such that \( AD:DC = 3:1 \)? Wait, no, \( AD = 9 \), so \( 9/DC=3/1 \), so \( DC = 3 \). Wait, let's do it properly.

By the Basic Proportionality Theorem (Thales' theorem), if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Here, \( DE\parallel BC \) (since the segments \( DE \) and \( BC \) have the same marking, indicating they are parallel). So, in \( \triangle ABC \), \( DE\parallel BC \), so \( \frac{AE}{EB}=\frac{AD}{DC} \).

We know that \( AE = 12 \), \( EB = 4 \), \( AD = 9 \). Let \( DC=x \). Then according to the theorem:

\( \frac{AE}{EB}=\frac{AD}{DC} \)

Substitute the known values:

\( \frac{12}{4}=\frac{9}{x} \)

Step2: Solve for \( x \)

Simplify \( \frac{12}{4}=3 \), so the equation becomes \( 3=\frac{9}{x} \).

Cross - multiply: \( 3x = 9 \)

Divide both sides by 3: \( x=\frac{9}{3}=3 \)

Answer:

3 units