A trapezoid is inscribed in a circle of radius 25, the bases of which are 30 and 40

A trapezoid is inscribed in a circle of radius 25, the bases of which are 30 and 40, and the center of the circle lies inside the trapezoid. Find the height of this trapezoid.

Since it is inscribed in a circle, this trapezoid is isosceles. Let’s build the radii ОА, ОВ, ОВ, ОD.

The BOC and AOD triangles are isosceles, and the OC and OM segments are their heights and medians.

In right-angled triangles BOK and AOM, we define the legs OK and OM by the Pythagorean theorem.

OK ^ 2 = OB ^ 2 – BK ^ 2 = 625 – 225 = 400.

OK = 20 cm.

ОМ ^ 2 = ОА ^ 2 – АМ ^ 2 = 625 – 400 = 225.

OM = 15 cm.

Then KM = BH = h = OK + OM = 20 + 15 = 35 cm.

Answer: The height of the trapezoid is 35 cm.