Question

3 fill in the blank 1 point find de. 13x - 37 b a 2x + 13 d c de = type your answer...

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the mid - points of two sides of a triangle) is half the length of the third side. Here, \(DE\) is the mid - segment of \(\triangle ABC\) (since \(AD = DC\) and \(CE=EB\)), so \(DE=\frac{1}{2}AB\).
We have \(DE = 2x + 13\) and \(AB=13x - 37\). Then \(2(2x + 13)=13x - 37\).

Step2: Solve the equation for \(x\)

Expand the left - hand side: \(4x+26 = 13x - 37\).
Subtract \(4x\) from both sides: \(26=13x - 4x-37\).
Simplify: \(26 = 9x-37\).
Add \(37\) to both sides: \(26 + 37=9x\), so \(63 = 9x\).
Divide both sides by \(9\): \(x = 7\).

Step3: Find the length of \(DE\)

Substitute \(x = 7\) into the expression for \(DE\): \(DE=2x + 13\).
\(DE=2\times7 + 13\).
\(DE=14 + 13\).

Answer:

\(27\)