Triangle DBC is isosceles with base DC. Its perimeter is 34 cm, side BD is 10 cm. Find the lengths of the line segments

Triangle DBC is isosceles with base DC. Its perimeter is 34 cm, side BD is 10 cm. Find the lengths of the line segments DN and BN (where N is the touch point of the inscribed circle with side DB).

Let us find the length of the base DC, since the triangle is isosceles:

| DC | = P – 2 * | BD | = 34 – 20 = 14 cm.

We denote the point of intersection of the height dropped from the vertex B to the side DC through M, the center of the inscribed circle through O. Then, by the Pythagorean theorem:

| BM | = √ (| BD | ^ 2 – | DC | ^ 2/4) = √ (100 – 49) = √51.

Using the property of medians, we get:

| OM | = 1/3 | BM |;

| OB | = 2/3 | BM |.

Since | OM | = r = | ON |, r is the radius.

Then:

| BN | = √ | OB | ^ 2 – | OM | ^ 2 = 1 / √3 * | BM | = √51 / √3 = √17.

| DN | = | BD | – | BN | = 10 – √17.