In triangle ABC, point G is the centroid. If the length of segment AG is equal to the length of side BC, then what is the measure of angle BGC in degrees?

Introduction / Context:
This is a higher level geometry question involving special points in a triangle and their relationships. The centroid is the point where the medians intersect. Here, a special condition is given: the segment from a vertex to the centroid has the same length as the opposite side. You are asked to determine the angle formed at the centroid between the two segments to the other vertices, which requires using properties of medians, centroids and analytic geometry or vector reasoning.

Given Data / Assumptions:

• Triangle ABC is given.
• G is the centroid of triangle ABC.
• Segment AG is a median from vertex A to side BC.
• The length AG is equal to the length of side BC.
• We must find the measure of angle BGC in degrees.
• Standard Euclidean geometry is assumed.

Concept / Approach:
The centroid of a triangle divides each median in the ratio 2 : 1, counted from the vertex to the midpoint of the opposite side. To handle the special length condition AG = BC, a convenient method is to use coordinate geometry. By placing the triangle symmetrically on the coordinate axes, we can express side lengths and the position of the centroid in terms of variables, then apply the condition AG = BC to fix the shape. Once coordinates of B, G and C are known, we can use the dot product to compute angle BGC.

Step-by-Step Solution:
Step 1: Place triangle ABC on a coordinate plane for simplicity. Step 2: Let B = (−a, 0) and C = (a, 0). Then BC is horizontal with length BC = 2a. Step 3: Place A at (0, h), so the triangle is symmetric about the y axis. Step 4: The centroid G is the average of the vertices: G = ((−a + a + 0) / 3, (0 + 0 + h) / 3) = (0, h / 3). Step 5: The length AG is the vertical distance from (0, h) to (0, h / 3), which is h − h / 3 = 2h / 3. Step 6: The side BC has length 2a. The given condition is AG = BC, so 2h / 3 = 2a. Step 7: Simplify to get h / 3 = a, hence h = 3a. Step 8: Substitute h = 3a into the coordinates. Then G = (0, a). Step 9: Now compute vectors GB and GC. GB = B − G = (−a − 0, 0 − a) = (−a, −a). GC = C − G = (a − 0, 0 − a) = (a, −a). Step 10: Use the dot product formula to find angle BGC. The dot product GB · GC = (−a)(a) + (−a)(−a) = −a^2 + a^2 = 0. Step 11: If the dot product of two non-zero vectors is zero, the angle between them is 90 degrees. Step 12: Therefore, angle BGC = 90°.

Verification / Alternative check:
You can choose a specific value of a, for example a = 1. Then B = (−1, 0), C = (1, 0), A = (0, 3), G = (0, 1). The vectors GB and GC are (−1, −1) and (1, −1) respectively. Plotting or calculating their slopes shows one has slope 1 and the other has slope −1, so they are perpendicular. The right angle at G confirms that angle BGC is 90 degrees, consistent with the general vector calculation.

Why Other Options Are Wrong:
45° and 60°: These would not make the vectors GB and GC perpendicular and would contradict the length condition AG = BC in the coordinate setup.
120° and 150°: These obtuse angles also do not arise from the perpendicular relationship implied by the dot product being zero.

Common Pitfalls:
The main difficulty lies in correctly using the centroid properties and the condition AG = BC. Some students forget that the centroid divides each median in a 2 : 1 ratio or try to apply only pure Euclidean geometry without a clear structure, leading to complicated constructions. Another common error is miscomputing vector dot products or misplacing points in the coordinate plane. Carefully defining coordinates and going step by step with algebra keeps the reasoning clear and reliable.

Final Answer:
The measure of angle BGC is 90°.