Question

The circles S1 and S2 intersect at points A and B. Additionally, the center of S1 is on S2, and the center of S2 is on S1. Point C is located on S1 such that AB=AC. Find the angle measures of △ABC.

a) 90°
b) 60°
c) 45°
d) 30°

Answer 1

Triangle ABC is an isosceles right triangle, with the right angle at point B. Hence, the angle measures at points A and C are both 45 degrees, corresponding to option c) 45°.

Step-by-step explanation:

The given problem describes two intersecting circles S1 and S2, with centers on each other and points A, B, and C such that AB = AC. From these properties, it can be deduced that triangle ABC is an isosceles triangle, and since the centers of the circles lie on each other, it implies that AB is not only the base of the isosceles triangle but also the diameter of both circles. Recall that a triangle inscribed in a circle where one side is a diameter forms a right triangle (Thales' theorem). Therefore, triangle ABC is a right isosceles triangle.

Since triangle ABC is isosceles and right-angled, the angles at A and C must be equal, and as it is a right-angled triangle, the right angle is at B. Each of the other two angles must then be 45°, as the sum of angles in any triangle is 180°. Hence, the angle measures of triangle ABC are 90°, 45°, and 45° at points B, A, and C respectively, corresponding to option c) 45°.