Difficulty: Easy
Correct Answer: 2√3 cm
Explanation:
Introduction:
This question focuses on two well known properties of triangles: the length of the median in an equilateral triangle and the position of the centroid along that median. It uses basic geometry facts to compute the distance from a vertex to the centroid.
Given Data / Assumptions:
Concept / Approach:
In any triangle, the centroid divides each median in the ratio 2 : 1, counting from the vertex to the midpoint of the opposite side. That is, the vertex to centroid segment is two thirds of the median. In an equilateral triangle, each median is also an altitude, and its length can be found using the 30°–60°–90° triangle relations.
Step-by-Step Solution:
Side length a = 6 cm. In an equilateral triangle, the height (and also the median) from any vertex is h = (√3 / 2) * a. So median length from vertex A, call it AM, is h = (√3 / 2) * 6 = 3√3 cm. The centroid G divides median AM in the ratio 2 : 1 from vertex to base. Therefore, AG = (2/3) * AM. AG = (2/3) * 3√3 = 2√3 cm.
Verification / Alternative check:
You can also coordinate the triangle with A at the top and base on the x axis, compute the centroid as the average of vertices, and then distance from A to G. This will also produce AG = 2√3 cm for side length 6, confirming the median based reasoning.
Why Other Options Are Wrong:
Values 2√2 cm and 3√2 cm relate to right triangles with 45° angles, not to equilateral triangles. The value 3√3 cm is the entire median length, not the distance from vertex to centroid. According to centroid properties, that distance must be two thirds of the median, giving 2√3 cm instead.
Common Pitfalls:
Confusing centroid with incenter or circumcenter can lead to wrong assumptions about distances. Another common error is to forget the 2 : 1 division of medians and instead use half. Remember that the centroid lies closer to the side than to the vertex.
Final Answer:
The length of AG is 2√3 cm.
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