Question

use the figure at the right. if ( be = 4x + 7 ) and ( df = 9x + 12 ), find df.

( df=square ) (simplify your answer.)

Explanation:

Step1: Use the mid - segment theorem

In a triangle, the mid - segment is half the length of the third side. Here, \(BE\) is a mid - segment of \(\triangle DAF\), so \(DF = 2BE\).
Since \(BE=4x + 7\) and \(DF = 9x+12\), we substitute into the equation \(DF = 2BE\).
We get \(9x + 12=2(4x + 7)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(9x + 12=8x + 14\).
Subtract \(8x\) from both sides: \(9x-8x+12=8x - 8x+14\), which simplifies to \(x+12 = 14\).
Subtract 12 from both sides: \(x=14 - 12\), so \(x = 2\).

Step3: Find the value of \(DF\)

Substitute \(x = 2\) into the expression for \(DF\).
\(DF=9x + 12\), then \(DF=9\times2+12\).
First, calculate \(9\times2=18\), then \(18 + 12=30\).

Answer:

\(30\)