Question

in a perpendicular angle prove that the square of hypotenuse is equal to sum of squares of other two sides use the above theorem to prove the following if angle abc is an equilateral angle and ad perpendicular bc then prove that ad² = 3dc²​

Answer 1

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

We have,

△ADB∼△ABC. (by AA similarity)

Therefore, AD/AB=AB/AC

(In similar Triangles corresponding sides are proportional)

AB²=AD×AC……..(1)

Also, △BDC∼△ABC

Therefore, CD/BC=BC/AC

(in similar Triangles corresponding sides are proportional)

Or, BC²=CD×AC……..(2)

Adding the equations (1) and (2) we get,

AB²+BC²=AD×AC+CD×AC

AB²+BC²=AC(AD+CD)

( From the figure AD+CD=AC)

AB²+BC²=AC.AC

Therefore, AC²=AB²+BC²

This theorem is known as Pythagoras theorem...