Question

In $\triangle ABC$, the acute angles are $\angle B=60^\circ$ and $\angle C=30^\circ$. The centroid of $\triangle ABC$ is $G$. If $AB=1,$ what is $AG?$

Answer 1

To calculate AG, we realize that ABC is a right triangle with AB = 1, and by the properties of a 30-60-90 triangle, the hypotenuse AC is twice AB. Therefore, AC = 2, and since the centroid divides the median in a 2:1 ratio, AG is 2/3 of AC, which equals 4/3 or approximately 1.33.

Step-by-step explanation:

The student has asked to calculate the length of AG, which is the segment from vertex A to the centroid G of triangle ABC, given that AB = 1 and the acute angles ∠B = 60° and ∠C = 30°.

In any triangle, the centroid divides the medians in the ratio 2:1. As triangle ABC is a right triangle due to the sum of angles ∠B and ∠C, we know that AC is the hypotenuse. By the properties of a 30-60-90 triangle,

AC would be twice the length of side AB, the side opposite the 30° angle. Therefore, AC = 2.

Given the ratio in which centroid divides the median, AG = 2/3 × AC,

giving us AG = 2/3 × 2 = 4/3 or approximately 1.33.