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

OR

State and prove the Pythagoras theorem.

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 ADE

Area of Δ BDE

So

… (2)

Again, area of Δ ADE

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.

⇒ Δ ADB ~ Δ ABC

So,

Or AD × AC = AB² … (1)

Also, Δ BDC ~ Δ ABC

So,

Or CD × AC = BC² … (2)

Adding (1) and (2),

AD × AC + CD × AC = AB² + BC²

⇒ AC (AD + CD) = AB² + BC²

⇒ AC² = AB² + BC²

Hence, proved.