# If a line is draw

Given: In Δ ABC, AB which intersects AB and AC at D and F respectively.

To Prove:

Construction:

Join B, E and C, D and then draw DM AC and EN AB.

Proof:

Area of Δ BDE

So

… (2)

Area of Δ CDE

So

…(3)

Observe that Δ BDE and Δ CDE are on the same base DE and between same parallels BC and DE.

So, ar(Δ BDE) = ar(Δ CDE) … (3)

From (1), (2) and (3), we have

Hence, proved.

OR

Statement: In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

Given: A right triangle ABC right angled at B.

To Prove: AC2 = AB2 + BC2

Construction: Draw BD AC

Proof:

We know that if a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

So,

Or AD × AC = AB2 … (1)

Also, Δ BDC ~ Δ ABC

So,

Or CD × AC = BC2 … (2)

AD × AC + CD × AC = AB2 + BC2

AC (AD + CD) = AB2 + BC2

AC2 = AB2 + BC2

Hence, proved.

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