Question

what is the length of bc? from the markings on the diagram we can tell e is the midpoint of bc and is the midpoint of ac we can apply the theorem. ed = 1/2 ba substituting in the expressions for the lengths and solving for x, we get x = now, since be = x, then bc =

Explanation:

Step1: Identify mid - points

From the markings on the diagram, E is the midpoint of \(BC\) and D is the midpoint of \(AC\).

Step2: Apply the mid - segment theorem

The mid - segment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So, \(ED=\frac{1}{2}BA\).

Step3: Set up the equation

Given \(ED=x + 2\) and \(BA=4x-6\). Substituting into \(ED=\frac{1}{2}BA\), we get \(x + 2=\frac{1}{2}(4x-6)\).

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Step4: Calculate \(BC\)

Since \(E\) is the midpoint of \(BC\) and \(BE=x\), then \(BC = 2BE\). Substituting \(x = 4\), we get \(BC=2\times4=8\).

Answer:

D is the midpoint of AC; mid - segment; \(x = 4\); \(BC=8\)