The perimeters of two similar triangles are 30 cm and 24 cm. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.

Let the triangles be ∆ABC and ∆DEF.

Given ∆ABC ~ ∆DEF

Since, corresponding sides of similar triangle are proportional to each other.

So, $\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF}$

Adding numerator and denominator we get :

$\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF} \\[1em] = \dfrac{AB + BC + AC}{DE + EF + DF} \\[1em] = \dfrac{\text{Perimeter of ΔABC}}{\text{Perimeter of ΔDEF}}$

So,

$\dfrac{\text{Perimeter of ΔABC}}{\text{Perimeter of ΔDEF }} = \dfrac{AB}{DE} \\[1em] \dfrac{30}{24} = \dfrac{12}{DE} \\[1em] DE = \dfrac{24 \times 12}{30} \\[1em] DE = 9.6 \text{ cm}.$

Hence, the length of corresponding side of second triangle is 9.6 cm.