- State BPT theorem and prove it.

# Hint: To prove this theorem first we will join BE and CD. Then draw a line EL perpendicular to AB and line DM perpendicular to AC. Now we will find the ratio of area of ADE to DBE and ratio of area of ADE to ECD. Comparing the ratios we will get the final answer.

## Complete step-by-step answer:

Now, and being on the same base DE and between the same parallels DE and BC, we have, then we say that the basic proportionality theorem is proved.Basic proportionality theorem:

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio.

**Given:**

A in which and DE intersects AB and AC at D and E respectively.

**To prove that:**

**Construction:**Join BE and CD.

Draw and

**Proof:**

We have the

Therefore the ratio of these two is . . . . . . . . . . . . . . (1)

Similarly,

Therefore the ratio of these two is . . . . . . . . . . . .. . . (2)

Now, and being on the same base DE and between the same parallels DE and BC, we have,

. . . . . . . . . . . (3)

From equations 1, 2, 3 we can conclude that

Hence we can say that the basic proportionality theorem is proved.

Note: The formula for area of the triangle is given by where b, h are base and height respectively. If two triangles are on the same base and between the same parallels then the area of those two triangles are equal.