Find the smaller diagonal in diamond ABCD if side AB is √19 and the large diagonal of BD is 2√10.

The diagonals of the rhombus BD and AC intersect at right angles and are halved at the point of intersection. Thus, the halves of the diagonals are the legs of a right-angled triangle, and the side of the rhombus AB is its hypotenuse. By the Pythagorean theorem, we can write:

(BD / 2) ^ 2 + (AC / 2) ^ 2 = AB ^ 2;

(AC / 2) ^ 2 = AB2 – (BD / 2) ^ 2 = (√19) ^ 2 – (2√10 / 2) ^ 2 = 19 – 10 = 9 = 32;

AC / 2 = 3;

AC = 3 * 2 = 6 – the required smaller diagonal of the given rhombus.