The perimeter of an isosceles triangle is 32 cm, and one of its sides is 12 cm. Find the lengths of the other two sides of the triangle. How many solutions does the problem have?
The solution of the problem.
The perimeter of a triangle (P) is the sum of the lengths of all its sides:
P = a + b + c, where a, b, c are the sides of the triangle.
In an isosceles triangle, the sides are equal.
Suppose that a = b = 12 cm, then P = 12 + 12 + s.
Knowing that the perimeter of the triangle is 32 cm, we find the third side of the triangle:
32 = 12 + 12 + s,
32 = 24 + s,
c = 32 – 24,
c = 8.
This means that the base of an isosceles triangle is 8 cm.
Since the problem statement does not say which side is 12 cm, we can assume that the base is 12 cm.Then the sides will be equal:
32 = a + a + 12,
32 = 2a + 12,
2a = 32 – 12,
2a = 20,
a = 20: 2,
a = 10.
We get that if the base of an isosceles triangle is 12 cm, then its sides are 10 cm.
Answer: the problem has two solutions.
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