Question

1. If the three sides of a right-angled triangle be in continued proportion, the hypotenuse is divided in extreme and mean ratio by the perpendicular from the right angle on the hypotenuse.

Answer 1

Because the sides of the right triangle are in continued proportion, and let a, b and c be the sides then b^2 = ac.
Also according to Pyhtagorean theorem, c^2=a^2+b^2
c^2-a^2=b^2

Hypotenuse AC is divided as AD and DC.

Since the ratios are the same, proved that the hypotenuse is divided in the extreme and mean ratio.