Three Dimensional Geometry

Given:

Isosceles triangle ABC with AB = AC = 15 and BC = 10.

O is the center of the inscribed circle.

P is the midpoint of BC.

We want to find the length of OP.

Step 1: Draw a diagram.

Draw isosceles triangle ABC with AB = AC = 15 and BC = 10. Draw the inscribed circle with center O. Since ABC is isosceles, the altitude from A to BC will bisect BC at point P. Thus, AP is the altitude and median to BC. Also, O will lie on AP. OP is the radius (r) of the inscribed circle.

Step 2: Find the length of AP.

Triangle ABP is a right-angled triangle with AB = 15 and BP = BC/2 = 10/2 = 5.

Using the Pythagorean theorem:

Step 3: Find the semi-perimeter (s) of triangle ABC.

Step 4: Find the area of triangle ABC.

Step 5: Find the inradius (r) of the inscribed circle.

The area of a triangle can also be expressed as , where r is the inradius and s is the semi-perimeter.

Since OP is the radius of the inscribed circle, OP = r.

Therefore, OP = units.