# A circle is inscribed in the rhombus. The point of tangency divides the side of the rhombus into segments

A circle is inscribed in the rhombus. The point of tangency divides the side of the rhombus into segments equal to 1 cm and 14 cm. What is the diameter of the circle?

Since the point G is the point of tangency, the segment OG is perpendicular to AB, and therefore OG is the height of the right-angled triangle ABO.

Consider a right-angled triangle ABO. Let the leg OA = X cm, and the leg OB = Y cm.

Then, by the Pythagorean theorem, AB ^ 2 = X ^ 2 + Y ^ 2 = (14 + 1) ^ 2 = 225. (1).

In a right-angled triangle AGO AG ^ 2 = X ^ 2 – OG ^ 2 = 14 ^ 2 = 196.

In a right-angled triangle BGO BG ^ 2 = Y ^ 2 – OG ^ 2 = 1 ^ 2 = 1.

Let’s add the last two equations.

X ^ 2 + Y ^ 2 – 2 * OG ^ 2 = 197.

X ^ 2 + Y ^ 2 = 197 + 2 * OG ^ 2. (2).

Let us subtract the second from the first equation.

X ^ 2 + Y ^ 2 – X ^ 2 – Y ^ 2 = 225 – 197 – 2 * OG ^ 2.

2 * OG ^ 2 = 28.

OG ^ 2 = 28/2 = 14.

OG = R = √14 cm 