Question

given: \\( \triangle abc \\) prove: the medians of \\( \triangle abc \\) are concurrent. proof: 1. the vertices of \\( \triangle abc \\) are unique points: \\( a(x_1,y_1) \\), \\( b(x_2,y_2) \\), and \\( c(x_3,y_3) \\). 2. use rigid transformations to transform \\( \triangle abc \\) into \\( \triangle arc \\). what is step 10 in this proof? a. statement: point \\( p \\) lies on line \\( ae \\) and line \\( bf \\). reason: algebra b. statement: all three lines share point \\( p \\). reason: definition of midpoint c. statement: point \\( p \\) lies on line \\( cd \\). reason: the coordinates of \\( p \\) satisfy the equation of line \\( cd \\). d. statement: two of the three medians share point \\( p \\). reason: using point - slope formula e. statement: line \\( ae \\) and line \\( bf \\) are concurrent. reason: definition of concurrence

Explanation:

Step1: Analyze the proof process

In a proof that medians of a triangle are concurrent, the final step is to state the concurrency.

Step2: Check the options

• Option A: Just stating a point lies on two lines is not the final concurrency statement.
• Option B: All three lines share a point \(P\) is a step towards concurrency but not the formal definition.
• Option C: Coordinates satisfying an equation is a calculation step.
• Option D: Two medians sharing a point \(P\) is an intermediate step.
• Option E: The definition of concurrency (three or more lines intersecting at a single point) is the final step.

Answer:

E. Statement line \(AE\) and line \(BF\) are concurrent. Reason: Definition of concurrence