In a trapezium, one diagonal divides the other diagonal in the ratio 2 : 9. If the length of the larger of the two parallel sides is 45 cm, what is the length, in centimetres, of the other parallel side?

Introduction / Context:
This question relies on a special property of trapeziums involving diagonals. When diagonals intersect, they generally divide each other in a ratio that is related to the lengths of the parallel sides (bases). Understanding this relationship allows us to move from information about the diagonals to information about the sides, which is a common technique in exam geometry questions.

Given Data / Assumptions:

• We have a trapezium with two parallel sides (bases).
• One diagonal divides the other diagonal in the ratio 2 : 9.
• The larger parallel side has length 45 cm.
• We must find the length of the smaller parallel side.
• The usual geometric properties of trapeziums and their diagonals are assumed to hold.

Concept / Approach:
In a trapezium, the diagonals intersect each other and divide each other internally in a ratio equal to the ratio of the lengths of the parallel sides. If the parallel sides have lengths a and b and one diagonal divides the other in the ratio a : b, then the ratio of the parallel sides is the same as the ratio of these segments. Hence, knowing the ratio in which one diagonal is divided allows us to determine the ratio of the bases. Here, the ratio is 2 : 9. The larger base corresponds to the larger segment (9 parts), and the smaller base corresponds to the smaller segment (2 parts).

Step-by-Step Solution:
Step 1: Let the lengths of the two parallel sides be L (larger) and S (smaller). Step 2: The problem states that one diagonal divides the other in the ratio 2 : 9. Step 3: By the property of trapezium diagonals, the ratio of the parallel sides equals the ratio in which the diagonals divide one another. Thus, L : S = 9 : 2, because the larger base corresponds to the larger segment (9 parts), and the smaller base corresponds to 2 parts. Step 4: We are given that L = 45 cm. So, 45 : S = 9 : 2. Step 5: Express S using proportion. 45 / S = 9 / 2 ⇒ S = (45 * 2) / 9 = 90 / 9 = 10 cm.

Verification / Alternative check:
To check, note that if S = 10 cm and L = 45 cm, then L : S = 45 : 10 = 9 : 2, which matches the diagonal division ratio 9 : 2. This is consistent with the trapezium diagonal property, confirming that 10 cm is the correct length for the smaller base.

Why Other Options Are Wrong:
Option 2: 9 cm would give a base ratio 45 : 9 = 5 : 1, which does not match the given 9 : 2 ratio from the diagonals. Option 3: 15 cm gives a ratio 45 : 15 = 3 : 1, again not equal to 9 : 2. Option 4: 18 cm gives 45 : 18 = 5 : 2, which still does not match 9 : 2. Option 5: None of these is incorrect because 10 cm is listed and is the only value that satisfies the required ratio.

Common Pitfalls:
Learners sometimes invert the ratio and incorrectly assume that the smaller base corresponds to the larger ratio part in the diagonal division. Another error is to assume that the ratio 2 : 9 directly gives the lengths of the bases without relating it to the given 45 cm. It is also easy to set up the proportion incorrectly, for example by writing 45 / S = 2 / 9 instead of 9 / 2. Carefully matching larger with larger and smaller with smaller in the ratio and double checking the proportion setup helps avoid mistakes.

Final Answer:
The length of the other parallel side is 10 cm.