Question

in the diagram below of triangle abc, d is the midpoint of \\( \overline { a c } \\) and e is the midpoint of \\( \overline { b c } \\). if \\( d e = 7 x - 12 \\), and \\( a b = - 6 x + 36 \\), what is the measure of \\( \overline { a b } \\)?

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (\(DE\)) of a triangle is half the length of the side parallel to it (\(AB\)). So, \(AB = 2DE\).

Step2: Substitute the given expressions

Substitute \(DE=7x - 12\) and \(AB=-6x + 36\) into \(AB = 2DE\). We get \(-6x+36=2(7x - 12)\).

Step3: Solve the equation

Expand the right - hand side: \(-6x + 36=14x-24\).
Add \(6x\) to both sides: \(36=14x + 6x-24\), so \(36=20x-24\).
Add \(24\) to both sides: \(36 + 24=20x\), \(60 = 20x\).
Divide both sides by \(20\): \(x = 3\).

Step4: Find the length of \(AB\)

Substitute \(x = 3\) into \(AB=-6x + 36\). Then \(AB=-6\times3+36\).
Calculate: \(AB=-18 + 36=18\).

Answer:

\(18\)