SOLUTION: The triangle is isosceles and the perimeter equals 54 cm. The height is 3*sqrt(21). Find the lengths of the sides and the area of the triangle. Express your answers in simplest rad

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Question 939227: The triangle is isosceles and the perimeter equals 54 cm. The height is 3*sqrt(21). Find the lengths of the sides and the area of the triangle. Express your answers in simplest radical form.
Answer by TimothyLamb(4379) (Show Source):
You can put this solution on YOUR website!
definition of isosceles triangle: a triangle with two equal sides
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the problem doesn't say which height is being talked about, so let's assume it's the height from the non-equal side (the base b), to the point where the two equal sides meet:
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perimeter = 54 = 2s + b
2s = 54 - b
s = (54 - b)/2
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length of the equal sides:
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s = sqrt( hh + bb/4 )
(54 - b)/2 = sqrt( hh + bb/4 )
(54 - b)(54 - b)/4 = hh + bb/4
54*54 - 108b + bb = 4hh + bb
2916 - 108b = 4hh
108b = 2916 - 4hh
b = (2916 - 4hh)/108
b = (2916 - 4* 3*sqrt(21) * 3*sqrt(21) )/108
b = (2916 - 4*3*3*sqrt(21)*sqrt(21) )/108
b = (2916 - 4*3*3*21 )/108
b = 20 cm
s = (54 - b)/2
s = (54 - 20)/2
s = 17 cm
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area:
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a = bh/2
a = 20*3*sqrt(21)/2
a = 30*sqrt(21) sq.cm
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