In a parallelogram PQRS, the lengths of the sides are 8 cm and 12 cm, and one of its diagonals has length 10 cm. What is the approximate length (in centimetres) of the other diagonal?

Introduction:
This question is about a parallelogram and the relationship between its sides and diagonals. For any parallelogram with sides of lengths a and b and diagonals of lengths d1 and d2, there is an important formula connecting them. Here we are given the side lengths and one diagonal, and we are asked to find the length of the other diagonal. This is a typical application of the parallelogram diagonal identity.

Given Data / Assumptions:

• Parallelogram PQRS has adjacent sides of lengths 8 cm and 12 cm. • One diagonal (say PR) has length 10 cm. • Let the other diagonal (QS) have length d cm. • We need to find the approximate value of d.

Concept / Approach:
For a parallelogram with side lengths a and b and diagonals d1 and d2, there is a standard relation: d1^2 + d2^2 = 2(a^2 + b^2). This formula is derived from vector geometry or from splitting the parallelogram into congruent triangles. We can use this identity by taking a = 8, b = 12, d1 = 10, and then solving for d2. Finally, we take the square root to obtain the length of the second diagonal, and then choose the closest option.

Step-by-Step Solution:
Step 1: Let a = 8 cm, b = 12 cm, d1 = 10 cm, and d2 = d cm. Step 2: Use the parallelogram diagonal relation. d1^2 + d2^2 = 2(a^2 + b^2). Step 3: Compute the right-hand side. a^2 = 8^2 = 64. b^2 = 12^2 = 144. a^2 + b^2 = 64 + 144 = 208. 2(a^2 + b^2) = 2 * 208 = 416. Step 4: Substitute into the formula. 10^2 + d^2 = 416. So, 100 + d^2 = 416. Step 5: Solve for d^2. d^2 = 416 - 100 = 316. Step 6: Find d by taking the square root. d = √316. Step 7: Approximate √316. Since 17^2 = 289 and 18^2 = 324, √316 is between 17 and 18, closer to 18. A more accurate value is d ≈ 17.78 cm.

Verification / Alternative check:
We can check the reasonableness: for sides 8 cm and 12 cm, the diagonals should both be larger than the shorter side but smaller than the sum of the two sides. The computed value d ≈ 17.78 cm is indeed greater than 12 and less than 20, which is consistent. Also, if we square 17.8, we get approximately 316.84, very close to 316, which confirms that 17.8 cm is a good approximation.

Why Other Options Are Wrong:
Option 17.5 cm: Squaring 17.5 gives 306.25, which is significantly less than 316. Option 17 cm: 17^2 = 289, which is far from 316. Option 18 cm: 18^2 = 324, which is slightly more than 316, but 17.8 cm is a closer match to √316. Option 16 cm: 16^2 = 256, which is much too small compared to the required 316.

Common Pitfalls:
Common errors include misremembering or misapplying the diagonal formula, mixing it up with a formula for rectangles, or trying to use the cosine rule without clearly identifying the included angle. Another mistake is approximating √316 too roughly as 18 without checking how far off that is. Knowing simple square numbers and interpolation between them helps in making a better approximation.

Final Answer:
The approximate length of the other diagonal is 17.8 cm.