The height of an equilateral triangle is 15, determine at what distance from its sides is the intersection point of its bisectors.
By condition, triangle ABC is equilateral, its height BH is also the bisector and median of the triangle. Let’s draw two medians, AK and CP. Their intersection point O divides the BH height in a ratio of 2/1 starting from the top B.
BО / BО = 2/1.
Let BO = 2 * X cm, then X cm, and 2 * X + X = 15.
3 * X = 15.
X = 15/3 = 5 cm.
BО = 2 * 5 = 10 cm.
OH = 15 – 10 = 5 cm.
In an equilateral triangle, point O is the center of both the inscribed and the circumscribed circle, then OP = OH = OK and are equal to the radius of the inscribed circle.
OP = OH = OK = 5 cm.
Answer: The distance from the point of intersection of the bisectors to the sides is 5 cm.
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