Question

D and E are points on the sides AB and AC respectively of $\Delta ABC$ . For the following case, state whether $DE\parallel BC$:
AB = 5.6 cm, AD = 1.4 cm, AC = 9.6 cm and EC = 2.4 cm
A. No
B. Yes
C. Ambiguous
D. Can’t say

Answer 1

Hint: As we know that there are many theorems for solving the problems related to triangles. We will prove this question using the ‘Triangle Proportionality Theorem’ which states that if a line is parallel to one side of a triangle intersects the other two sides, and then it divides those sides into proportional segments.
This is also called the “Side-Splitting Theorem” because the mid-segment splits the sides that it intersects. In $\Delta ABC$ with $DE\parallel BC$
$\therefore \dfrac{{AE}}{{EC}} = \dfrac{{AD}}{{DB}}$

Complete step by step solution:
Given that,
AB = 5.6 cm
AD = 1.4 cm
AC = 9.6 cm
EC = 2.4 cm


From figure,
AC = AE + EC
Let AE be x,
\[\begin{array}{*{20}{l}}
  {9.6{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}2.4} \\
  {x{\text{ }} = {\text{ }}7.2{\text{ }}cm}
\end{array}\]
and AB = AD + DB
Let DB be y,
\[\begin{array}{*{20}{l}}
  {5.6{\text{ }} = {\text{ }}1.4{\text{ }} + {\text{ }}y} \\
  {y{\text{ }} = {\text{ }}4.2{\text{ }}cm}
\end{array}\]
To check whether$DE\parallel BC$, we will use Triangle Proportionality Theorem
$\therefore \dfrac{{AE}}{{EC}} = \dfrac{{AD}}{{DB}}$
$
  \dfrac{{7.2}}{{2.4}} = \dfrac{{1.4}}{{4.2}} \\
  \dfrac{3}{1} \ne \dfrac{1}{3} \\
$

Hence, the ratios are not equal therefore DE is not parallel to BC.

∴Option (A) is correct.

Note: Students must keep in mind that we can use the Side-Splitter Theorem only for the four segments on the split sides of the triangle. The side splitter theorem is a natural extension of similarity ratio, and it happens any time that a pair of parallel lines intersect a triangle. Do not use it for the parallel sides, which are in a different ratio. The Side-Splitter only applies to the intercepted sides. It does not apply to the bottoms.

When you move the parallel line, you are changing the proportion between the upper and lower segments.
When you move the point, the segments may get longer or shorter, but the proportion stays the same.