The perimeter of a triangle is 28 cm. One of its sides is 8 cm. Write all the sides of the possible isosceles triangles with these measurements.
Sides of the triangles = (8 cm, 8 cm, 12 cm) or (10 cm, 10 cm, 8 cm)
Idea: In an isosceles triangle, two sides are equal. The perimeter is the sum of all three sides.
Given:
Perimeter = (28 ext{cm})
One side = (8 ext{cm})
We have two possible cases:
Case 1: The two equal sides are both (8 ext{cm})
Sum of the two equal sides = (8 + 8 = 16 ext{cm})
Third side = (28 - 16 = 12 ext{cm})
So the sides are (8 ext{cm}, 8 ext{cm}, 12 ext{cm}).
Check triangle inequality:
(8 + 8 > 12)
(8 + 12 > 8)
(8 + 12 > 8)
All true, so this triangle is possible.
Case 2: The side (8 ext{cm}) is the base (the unequal side)
Let each equal side be (x ext{cm}).
Perimeter: (x + x + 8 = 28)
(2x = 28 - 8)
(2x = 20)
(x = 10)
So the sides are (10 ext{cm}, 10 ext{cm}, 8 ext{cm}).
Check triangle inequality:
(10 + 10 > 8)
(10 + 8 > 10)
(10 + 8 > 10)
All true, so this triangle is possible.
Answer: The possible isosceles triangles are (8 ext{cm}, 8 ext{cm}, 12 ext{cm}) and (10 ext{cm}, 10 ext{cm}, 8 ext{cm}).