Part 1)
we know that
The Centroid of a Triangle is the centre of the triangle that can be calculated as the point of intersection of all the three medians of a triangle.
The Centroid divides each median into two segments whose lengths are in the ratio [tex]2:1[/tex]
so
[tex]VZ=\frac{1}{3}RV[/tex]
[tex]RV=3VZ[/tex]
we have
[tex]VZ=6\ in[/tex]
substitute
[tex]RV=3*6=18\ in[/tex]
Find RZ
[tex]RZ=\frac{2}{3}RV[/tex]
[tex]RZ=\frac{2}{3}*18=12\ in[/tex]
therefore
the answer Part 1) is the option C
[tex]12\ in[/tex]
Part 2)
Statements
case A) ∠BEC is an exterior angle
The statement is False
Because, ∠BEC is a internal angle
case B) ∠DEC is an exterior angle.
we know that
An exterior angle is formed by one side of a triangle and the extension of another side
therefore
The statement is True
case C) ∠ABE and ∠EBC are supplementary angles.
we know that
∠ABE+∠EBC=[tex]180\°[/tex] -------> by supplementary angles
therefore
The statement is True
case D) ∠BCF and ∠BEC are supplementary angles
The statement is False
Because
the only way that is true is that the triangle BEC is isosceles and that the ∠BEC is equal to the ∠BCE
case E) ∠BEC is a remote interior angle to exterior F.∠BCF
we know that
Remote interior angles are the interior angles of a triangle that are not adjacent to a given angle. Each interior angle of a triangle has two remote exterior angles.
In this problem ∠BEC has two remote exterior angles (∠BCF and ∠EBA)
therefore
The statement is True
