In triangle ABC, X and Y are the midpoints of BC and AC, respectively. The perpendicular bisectors of BC and AC intersect at a point O inside the triangle. If angle C = 47 degrees, then find the measure of angle XOY?
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To find the measure of angle XOY, we need to consider the properties of perpendicular bisectors and midpoints in a triangle.
Given that X and Y are the midpoints of BC and AC respectively, that means BX = CX and AY = CY. This is because the midpoints of a side of a triangle divide the side into two congruent segments.
Let's label the point of intersection of the perpendicular bisectors as O. Since O lies on the perpendicular bisector of BC, it must be equidistant from B and C. Similarly, since O lies on the perpendicular bisector of AC, it must be equidistant from A and C.
From these properties, we can conclude that O is the circumcenter of triangle ABC. The circumcenter is the point where the perpendicular bisectors of a triangle intersect.
Since the circumcenter is equidistant from the vertices of the triangle, the angles subtended at the circumcenter by these vertices are twice the corresponding angles of the triangle.
Therefore, angle XOY, which is subtended at the circumcenter O, will be twice the measure of angle ACB.
Since angle C = 47 degrees, angle ACB would be half of that, which is 47/2 = 23.5 degrees.
Therefore, angle XOY will be twice that, which is 2 * 23.5 = 47 degrees.
So, the measure of angle XOY is 47 degrees.