Through the center of a circle inscribed in a regular triangle with side 12, a straight line is drawn parallel to one side of the triangle.

| January 16, 2021 | education

Through the center of a circle inscribed in a regular triangle with side 12, a straight line is drawn parallel to one side of the triangle. Find the length of the line segment between the other two sides of the triangle.

Since triangle ABC is correct, its height BH is also its median, then AH = CH = AC / 2 = 12/2 = 6 cm.
Let us determine the length of the ВН leg from the right-angled triangle ABН.
BH2 = AB2 – AH2 = 122 – 62 = 144 – 36 = 108.
BH = 6 * √3 cm.
According to the property of the heights of a regular triangle, the kidney of their intersection divides the height into segments in a ratio of 2/1, starting from the apex. ВO / НO = 2 / 1. ВO = 2 * НO.
ВН = ВO + HO = 2 * HO + HO = 3 * HO = 6 * √3 cm.
НО = 2 * √3 cm.
ВO = 4 * √3 cm.
Triangles AВН and KOВ are similar in acute angle, then AН / KO = ВН / ВO.
6 / KO = 6 * √3 / 4 * √3.
6 * KO = 6 * 4.
KO = 4 cm.
Then KM = 2 * KO = 2 * 4 = 8 cm.
Answer: The length of the KM segment is 8 cm.



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