Question

gh || ik. find jk.
jk =

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{GH} \parallel \overline{IK} \), triangles \( \triangle JIK \) and \( \triangle JHG \) are similar by the Basic Proportionality Theorem (Thales' theorem).

Step2: Set Up Proportion

Let \( JK = x \). The ratio of corresponding sides in similar triangles is equal. So, \( \frac{JK}{JG}=\frac{JI}{JH} \). We know \( JI = 24 \), \( IH = 48 \), so \( JH=24 + 48=72 \), and \( JG=x + 24 \). Substituting values: \( \frac{x}{x + 24}=\frac{24}{72} \).

Step3: Solve the Proportion

Simplify \( \frac{24}{72}=\frac{1}{3} \), so the equation becomes \( \frac{x}{x + 24}=\frac{1}{3} \). Cross - multiply: \( 3x=x + 24 \). Subtract \( x \) from both sides: \( 3x-x=x + 24-x \), which gives \( 2x = 24 \). Divide both sides by 2: \( x=\frac{24}{2}=12 \).

Answer:

\( JK = 12 \)