The bisector of an isosceles triangle divides the height from the base into segments 20 cm and 16 cm long.

| January 13, 2021 | education

The bisector of an isosceles triangle divides the height from the base into segments 20 cm and 16 cm long. Find the perimeter of the triangle.

Determine the length of the HВ height.
ВН = ВK + KН = 20 + 16 = 36 cm.
Since AM is the bisector of the angle BAC, then AK is also the bisector of the angle ВAН.
Then, by the property of the angle bisector: AB / BK = AН / KН.
AB / 20 = AН / 16.
AB / AH = 20/16 = 5/4.
Let the length of the segment AH = 4 * X cm, then AB = 5 * X cm.
From the right-angled triangle ABН, according to the Pythagorean theorem:
BH ^ 2 = AB ^ 2 – AH ^ 2.
1296 = 25 * X ^ 2 – 16 * X ^ 2.
9 * X ^ 2 = 1296.
X ^ 2 = 144.
X = 12 cm.
Then AB = BC = 5 * 12 = 60 cm.
AH = 4 * 12 = 48 cm.
Since ABC is isosceles, then CH = AH = 48 cm.Then AC = 2 * 48 = 96 cm.
Ravs = AB + BC + AC = 60 + 60 + 96 = 216 cm.
Answer: The perimeter of the triangle is 216 cm.



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