# E is the mid-point of a median AD of ∆ABC and BE is produced to meet AC at F. Show that AF = 1/3 AC.

**Solution:**

Given, ABC is a triangle

E is the midpoint of a median AD

BE is produced to meet AC at F

We have to show that AF = 1/3 AC

Draw DP parallel to EF

Considering triangle ADP,

E is the midpoint of AD

EF || DP

By converse of __midpoint theorem__,

F is the midpoint of AP.

Considering triangle FBC,

D is the midpoint of BC

DP || BF

By converse of midpoint theorem,

P is the midpoint of FC

So, AF = FP = PC

Therefore, AF = 1/3 AC

**✦ Try This: **In the adjacent figure, ar(PEA) = ar(PAC) and ar(RPC) = ar(KEA) show that quadrilateral PARK and PACE are trapezium.

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 10**

## E is the mid-point of a median AD of ∆ABC and BE is produced to meet AC at F. Show that AF = 1/3 AC.

**Summary:**

E is the mid-point of a median AD of ∆ABC and BE is produced to meet AC at F. It is shown that AF = 1/3 AC by the converse of midpoint theorem which states that if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side

**☛ Related Questions:**

- Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is . . . .
- E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Pro . . . .
- Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle

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