In two similar triangles, triangle GHI and triangle KLM, the ratio of their perimeters is given as Perimeter of triangle GHI : Perimeter of triangle KLM = 1 : 4. If the length of side GH in triangle GHI is 2 cm, what is the length, in centimetres, of the corresponding side KL in triangle KLM?

Introduction / Context:
This problem checks your understanding of similarity of triangles and how perimeters and corresponding sides scale with the same ratio. Similar triangles are a core concept in geometry and are widely used in indirect measurement, trigonometry and many aptitude questions.

Given Data / Assumptions:

• Triangle GHI is similar to triangle KLM.

• Ratio of perimeters: Perimeter of GHI : Perimeter of KLM = 1 : 4.

• Length of side GH in triangle GHI = 2 cm.

• Side KL corresponds to side GH in the similar triangle KLM.

Concept / Approach:
For similar triangles, all corresponding linear dimensions (such as sides, medians, altitudes and perimeters) are in the same ratio, called the scale factor. If the perimeter of one triangle is multiplied by some factor to obtain the perimeter of another similar triangle, then each corresponding side is also multiplied by the same factor. Therefore, we can use the perimeter ratio directly as the side ratio between corresponding sides GH and KL.

Step-by-Step Solution:
Perimeter ratio (GHI : KLM) = 1 : 4. This means that every linear dimension of triangle KLM is 4 times the corresponding linear dimension of triangle GHI. Given GH = 2 cm in triangle GHI. Scale factor from triangle GHI to triangle KLM = 4. Therefore, KL = 4 * GH. KL = 4 * 2 cm = 8 cm.

Verification / Alternative check:
To verify, imagine that triangle GHI has sides 2 cm, 3 cm and 4 cm, for example. A similar triangle with perimeter four times larger would have sides 8 cm, 12 cm and 16 cm, all multiplied by the same factor 4. This confirms that any single corresponding side must also be multiplied by 4. Since GH is 2 cm, multiplying by 4 again gives KL = 8 cm, matching our earlier calculation.

Why Other Options Are Wrong:
The value 4 cm would correspond to a scale factor of 2, not 4, so it contradicts the given perimeter ratio. The values 16 cm and 32 cm are too large and would require scale factors of 8 and 16 respectively, which do not match the 1 : 4 ratio. Only 8 cm is consistent with the perimeter ratio and the rules of similarity.

Common Pitfalls:
One common mistake is to confuse area ratio with linear ratio. For similar figures, area grows with the square of the scale factor, but sides and perimeters grow linearly. Another error is to invert the ratio accidentally and divide by 4 instead of multiplying by 4. Always check whether you are going from the smaller figure to the larger figure or the other way around before applying the ratio.

Final Answer:
The length of the corresponding side KL in triangle KLM is 8 cm.