Let ABC be a triangle with\angle BAC = 90. A circle is tangent to the sides AB and AC at X and Y respectively, such that the points on the circle diametrically opposite X and Y both lie on the side BC. Given that AB = 6, find the area of the portion of the circle that lies outside the triangle.

qwertyzz Aug 8, 2020

#1**+3 **

Let the center of a circle be O and the midpoint of BC be M

Angle B = 45º

AM = sin( 45º) * 6

XY = 2/3 * AM (**Segment XY is a side of a square inscribed in a circle.)**

Area of an inscribed square A_{s} = XY^{2}

r = sin(45º) * XY

Area of a circle A_{c} = r^{2}pi

Area of a shaded segment **A = (A _{c} - A_{s}) / 4 = pi - 2**

Dragan Aug 8, 2020