Difficulty: Easy
Correct Answer: 84
Explanation:
Introduction:
This question checks whether you remember the area formula for a rhombus expressed in terms of its diagonals. A rhombus can be thought of as a special kite where the diagonals are perpendicular bisectors of each other. This gives a convenient formula for its area.
Given Data / Assumptions:
Concept / Approach:
For any rhombus (and in fact for any kite with perpendicular diagonals), the area can be calculated using the formula: Area = (1/2) * d₁ * d₂. This comes from the fact that the diagonals divide the rhombus into four right triangles of equal area, and their combined area equals half the product of the diagonals.
Step-by-Step Solution:
Given d₁ = 12 cm and d₂ = 14 cm. Use the formula: Area = (1/2) * d₁ * d₂. Substitute the values: Area = (1/2) * 12 * 14. Compute 12 * 14 = 168. Now multiply by 1/2: Area = 168 / 2 = 84 cm².
Verification / Alternative check:
If you draw the rhombus and its diagonals, you get four congruent right triangles, each with legs 6 cm and 7 cm (half of each diagonal). Area of one triangle is (1/2) * 6 * 7 = 21 cm². Multiply by 4 to get the rhombus area: 4 * 21 = 84 cm², confirming the formula based calculation.
Why Other Options Are Wrong:
The value 168 cm² is simply the product of the diagonals, without the necessary factor of 1/2. The values 42 cm² and 63 cm² are partial products that do not correspond to any standard area computation for rhombuses with these diagonals.
Common Pitfalls:
A typical mistake is to forget dividing by 2 and use d₁ * d₂ directly. Another is to mix formulas for rectangles or parallelograms with those of rhombuses. Remember that for rhombuses, diagonals and their perpendicular property are a powerful shortcut.
Final Answer:
The area of the rhombus is 84 cm².
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