In the figure, points
\(A\),
\(
B\), and
\(
C\) lie on
\(OX\), \(OY\) and \(OZ\) respectively. If \(AB||XY\) and \(AC||XZ\), Show that
\(BC∥YZ\).
Proof:
We know that, If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
In \(\Delta OXY\),
\(AB ||\)
\(\frac{OA}{AX} = \) - - - - (1)
And also in \(\Delta OXZ\),
\(AC ||\)
\(\frac{OC}{CZ} = \) - - - - (2)
From (1) and (2),
\(\frac{OC}{CZ} =\)
In \(\Delta OYZ\),
\(\frac{OC}{CZ} = \frac{OB}{BY}\)
By theorem,
Thus, \(BC||YZ\).
Answer variants:
\(XY\)
\(XZ\)
\(\frac{OB}{BY}\)
\(\frac{OA}{AX}\)