The height of the equilateral triangle ABC is 8.4 cm. Find the height of the triangle AOB drawn from the vertex O if the point O is equidistant from the sides of the triangle ABC.
By condition, triangle ABC is equilateral, its height BH is also the bisector and median of the triangle. By condition, point O is equidistant from the sides of the triangle, so point O is the intersection point of the medians, heights and bisectors of the triangle. Let’s draw two medians of the triangle AK and CP. Their intersection point O divides the BH height in a ratio of 2/1 starting from the top B.
BО / HО = 2/1.
Let BO = 2 * X cm, then X cm, and 2 * X + X = 8.4.
3 * X = 8.4.
X = 8.4 / 3 = 2.8 cm.
OH = 2.8 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 = 2.8 cm.
Answer: The length of the height of the ABO triangle is 2.8 cm.
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