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The perimeter of a triangle is 28cm. One of its sides is 8cm. Find all the sides of the possible isosceles triangles with these measurements.

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(10cm, 10cm, 8cm) and (8cm, 8cm, 12cm)

(11cm, 8cm, 8cm) and (8cm, 12cm, 12cm)

(10cm, 8cm, 8cm) and (8cm, 8cm, 11cm)

(11cm, 11cm, 8cm) and (8cm, 11cm, 11cm)

answer is A.

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Detailed Solution

Given that, perimeter of a triangle = 28 cm and one of its sides = 8cm.
Consider a triangle ABC.
Case 1-
If

AB = BC

then

AB = BC = 8 cm

.
Perimeter of

Δ ABC = 28 cm

\Rightarrow AB + BC + CA = 28 cm

\Rightarrow 8 + 8 + CA = 28

\Rightarrow 16 + CA = 28

\Rightarrow CA = 28 - 16

\Rightarrow CA = 12 cm

∴ The sides of isosceles triangle are 8 cm, 8 cm, 12 cm.
Case 2-
If

BC = CA

and

AB = 8 cm

.
Perimeter of

Δ

ABC = 28 cm

\Rightarrow AB + BC + CA = 28 cm

\Rightarrow AB + CA + CA = 28

\Rightarrow AB + 2 CA = 28

\Rightarrow 8 + 2 CA = 28

\Rightarrow 2 CA = 28 - 8

\Rightarrow 2 CA = 20

\Rightarrow CA = \frac{20}{2}

\Rightarrow CA = 10

\Rightarrow BC = CA = 10

∴ The sides of isosceles triangle are 8 cm,10 cm, 10 cm.
Therefore, the sides of all isosceles triangles are (8cm, 8cm, and 12cm) and (10cm, 10cm, and 8cm).
Hence, option (1) is correct.

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