A rhombus has perimeter 34 cm and one of its diagonals has length 8 cm. What is the length (in cm) of the other diagonal?

Introduction / Context:
This question is about a rhombus, a special quadrilateral in which all sides are equal. It uses the fact that the diagonals of a rhombus are perpendicular bisectors of each other. By combining this property with the given perimeter and one diagonal, you can find the length of the other diagonal using the Pythagoras theorem.

Given Data / Assumptions:

• The figure is a rhombus.
• Perimeter = 34 cm.
• One diagonal (say d1) has length 8 cm.
• Let the other diagonal be d2 (unknown).
• All sides of the rhombus are equal; let each side be s.
• The diagonals intersect at right angles and bisect each other.

Concept / Approach:
Because the perimeter is 34 cm and the rhombus has four equal sides, each side length s is 34 / 4. The diagonals cross at the centre and are perpendicular, so each half diagonal forms the legs of a right triangle whose hypotenuse is a side of the rhombus. Specifically: s² = (d1 / 2)² + (d2 / 2)² We can substitute the known values for s and d1 and solve for d2.

Step-by-Step Solution:
Step 1: Compute the side length s from the perimeter: s = 34 / 4 = 8.5 cm. Step 2: The given diagonal d1 = 8 cm, so half of it is d1 / 2 = 4 cm. Step 3: Let the other diagonal be d2; then half of it is d2 / 2. Step 4: Use the right triangle formed by half diagonals and a side: s² = (d1 / 2)² + (d2 / 2)². Step 5: Substitute s = 8.5 and d1 / 2 = 4: (8.5)² = 4² + (d2 / 2)². Step 6: Compute (8.5)² = 72.25 and 4² = 16. Step 7: So 72.25 = 16 + (d2 / 2)², giving (d2 / 2)² = 72.25 − 16 = 56.25. Step 8: Take the square root: d2 / 2 = √56.25 = 7.5. Step 9: Therefore d2 = 2 * 7.5 = 15 cm.

Verification / Alternative check:
We can verify by recomputing the side length from the diagonals: using d1 = 8 and d2 = 15, half diagonals are 4 and 7.5. Then the side length s should be √(4² + 7.5²) = √(16 + 56.25) = √72.25 = 8.5 cm. Multiplying by 4 gives perimeter 4 * 8.5 = 34 cm, which matches the given perimeter. This consistency confirms that d2 = 15 cm is correct.

Why Other Options Are Wrong:
7.5 cm is only half of the required diagonal. 30 cm and 22.5 cm would create sides that are much longer than 8.5 cm and lead to incorrect perimeter values. 8 cm is the length of the first diagonal and cannot be the second diagonal here without changing the given perimeter relationship and side length. None of these other values satisfy the right triangle relation between half diagonals and side length.

Common Pitfalls:
Some students mistakenly use the full diagonals directly in the Pythagoras relation without halving them, which leads to incorrect equations. Others forget that the diagonals of a rhombus are perpendicular, a property that is essential for forming the right triangle used in the calculation. Carefully halving the diagonals and using the side derived from the perimeter avoids these errors.

Final Answer:
The length of the other diagonal is 15 cm.