# The distance from the center of the inscribed circle in a right-angled triangle

The distance from the center of the inscribed circle in a right-angled triangle to the ends of the hypotenuse is √ 5 and √10. Find the length of the hypotenuse.

Since triangle ABC is rectangular, the sum of its acute angles is 90.

The center of the circle point O is the intersection point of the bisectors, then the sum of the angles (OAB + OBA) = 90/2 = 45.

Angle AOB = (180 – (OAB + OBA)) = (180 – 45) = 135.

In the triangle AOB, by the cosine theorem:

AB ^ 2 = ОА ^ 2 + ОВ ^ 2 – 2 * ОА * ОВ * Cos135 = 15 – 2 * √50 * (-√2 / 2) = 15 + 2 * 5 * √2 * √2 / 2 = 25 …

AB = 5 cm.

Answer: The length of the hypotenuse is 5 cm. One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.