# D and E are points on sides AB and AC respectively of ∆ ABC such that ar (DBC) = ar (EBC). Prove that DE || BC.

**Solution:**

If two triangles are on a common base and have equal areas, then they will lie between the same parallel lines.

Let's draw the given triangle ABC.

Given that ar (DBC) = ar (EBC)

Since ΔEBC and ΔDBC are lying on a common base BC and they also have equal areas, therefore, ΔEBC and ΔDBC will lie between the same parallel lines.

Hence, DE || BC proved.

**☛ Check: **Class 9 Maths NCERT Solutions Chapter 9

**Video Solution:**

## D and E are points on sides AB and AC respectively of ∆ ABC such that ar (DBC) = ar (EBC). Prove that DE || BC.

Maths NCERT Solutions Class 9 Chapter 9 Exercise 9.3 Question 7:

**Summary:**

If D and E are points on sides AB and AC respectively of such that ar (DBC) = ar (EBC), then DE||BC.

**☛ Related Questions:**

- XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar (ABE) = ar (ACF)
- The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see Fig. 9.26). Show that ar (ABCD) = ar (PBQR).
- Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).
- In Fig. 9.27, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that i) ar (ACB) = ar (ACF) ii) ar (AEDF) = ar (ABCDE)