The distance from the center of the inscribed circle in a right-angled triangle
| September 14, 2021 | education
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.
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