Question

1. find the value of x in the figure below if \\(\overline{re}\\) is parallel to \\(\overline{pm}\\). figure of a triangle with points a, r, p on one side and a, e, m on the other, with lengths 20 (ar), 25 (ae), 9 (em), and x (rp). \\(\overline{re}\\) is parallel to \\(\overline{pm}\\). \\(\bigcirc\\) 9 units \\(\bigcirc\\) 7.2 units \\(\bigcirc\\) 5 units \\(\bigcirc\\) 14.4 units

Explanation:

Step1: Apply Basic Proportionality Theorem

Since \( \overline{RE} \parallel \overline{PM} \), by the Basic Proportionality Theorem (Thales' theorem), the line parallel to one side of a triangle divides the other two sides proportionally. So, \( \frac{AR}{RP} = \frac{AE}{EM} \).
Given \( AR = 20 \), \( RP = x \), \( AE = 25 \), \( EM = 9 \). Substituting these values, we get \( \frac{20}{x} = \frac{25}{9} \).

Step2: Solve for \( x \)

Cross - multiply the proportion \( \frac{20}{x}=\frac{25}{9} \) to get \( 25x=20\times9 \).
Calculate \( 20\times9 = 180 \), so the equation becomes \( 25x = 180 \).
Then, \( x=\frac{180}{25}=\frac{36}{5} = 7.2 \).

Answer:

7.2 units