The angle at the apex of an isosceles triangle is 2α, and the radius of the circle inscribed in it is r.

| June 20, 2021 | education

The angle at the apex of an isosceles triangle is 2α, and the radius of the circle inscribed in it is r. Find the side of the triangle.

From the vertex A, we construct the height AH, which is also the median and the bisector, then BH = CH, the angle ABH = BAC / 2 = 2 * α / 2 = α0.

Let’s construct the height OM to the side AB, which is the radius of the inscribed circle.

In a right-angled triangle AOM tgα = OM / AM.

AM = OM / tgα = r / tgα cm.

Angle ABН= (90 – α). The ВН segment is the bisector of the AВН angle, the MВO angle = (90 – α) / 2 = (45 – α / 2).

In a right-angled triangle PTO tgOBM = OM / BM.

BM = OM / tan (45 – α / 2) = r / tan (45 – α / 2) cm.

Then AB = AM + BM = r / tanα + r / tan (45 – α / 2) = r * (1 / tanα + 1 / (tan (45 – α / 2)) = r * (ctgα + ctg (45 – α / 2)) cm.

Answer: The length of the side is r * (ctgα + ctg (45 – α / 2)) 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.