Question

XYZ is an equilateral triangle, inscribed in a circle. P is a point on the arc YZ such that X and P are on opposite sides of the chord YZ. Which of the following MUST always be true?

• A. XZ + YP = XY + PZ
• B. XP = XY
• C. XP + PZ = XY + YP
• D. XP = YP + PZ
• E. XP = XY + YZ

Answer 1

The Correct Option is D

P is a point on the arc YZ of the circle in which the equilateral triangle XYZ is inscribed. X and P are on opposite sides of the chord YZ, therefore P lies on the major arc YZ. In a circle, when a point is located on the arc opposite to a chord, the distance from the point (P) to the vertices of the chord (Y and Z) plus the distance from the other vertex (X) of the triangle to the point must equate the total length of the arc. Thus, the formula becomes:

XP = YP + PZ

This is derived from the properties of triangles and circles, where lines drawn from a point on a circle to its perimeter are related to the arc they subtend. Specifically, for the given configuration, P is part of the major arc, meaning that it lies within the triangle's circumcircle and opposite the chord YZ.

Therefore, the length XP is logically the sum of YP and PZ because P divides the arc YZ into two parts without considering the direct line segment YZ.