Question

use the figure at the right. if ( be = 2x + 5 ) and ( df = 5x + 3 ), find df.
( df=square ) (simplify your answer.)

Explanation:

Step1: Use the property of mid - segment in a triangle

In a triangle, if a line segment connects the mid - points of two sides of a triangle, then the length of this line segment is half of the length of the third side. Here, \(BE\) is the mid - segment of \(\triangle ADF\) (assuming \(B\) is the mid - point of \(AD\) and \(E\) is the mid - point of \(AF\)), so \(DF = 2BE\).

Step2: Substitute the given expressions

Since \(BE=2x + 5\) and \(DF = 5x+3\), and \(DF = 2BE\), we substitute \(BE\) into the equation: \(5x+3=2(2x + 5)\).

Step3: Solve the equation for \(x\)

Expand the right - hand side: \(5x+3 = 4x+10\).
Subtract \(4x\) from both sides: \(5x-4x+3=4x - 4x+10\), which gives \(x+3=10\).
Subtract 3 from both sides: \(x=10 - 3=7\).

Step4: Find the value of \(DF\)

Substitute \(x = 7\) into the expression for \(DF\): \(DF=5x+3\).
\(DF=5\times7+3=35 + 3=38\).

Answer:

\(38\)