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Quadrilaterals (Parallelogram, Rhombus, Trapezium)

Introduction

Quadrilaterals चार-भुजी polygons होते हैं जिनकी अलग-अलग properties होती हैं। aptitude और geometry-based exams में अक्सर ऐसे questions आते हैं जहाँ parallelograms, rhombuses और trapeziums जैसी figures के area, perimeter, और sides, diagonals या heights के relations निकाले जाते हैं।

इनके खास formulas और geometric properties समझने से problem-solving के दौरान जल्दी सही approach चुनने में मदद मिलती है।

Pattern: Quadrilaterals (Parallelogram, Rhombus, Trapezium)

Pattern

मुख्य idea यह है कि पहले quadrilateral का प्रकार पहचानें और फिर उसके area या diagonal वाले formula को सही तरीके से apply करें।

Key Formulas:

  • Parallelogram: Area = base × height; Perimeter = 2(a + b)
  • Rhombus: Area = ½ × d₁ × d₂; सभी sides बराबर होती हैं, diagonals right angles पर bisect होती हैं।
  • Trapezium: Area = ½ × (sum of parallel sides) × height

Step-by-Step Example

Question

किसी rhombus की diagonals 24 cm और 10 cm हैं। उसका area निकालें।

Solution

  1. Step 1: Rhombus के area वाला formula याद करें।

    Area = ½ × d₁ × d₂

  2. Step 2: दिए गए मान substitute करें।

    Area = ½ × 24 × 10

  3. Step 3: Calculate करें।

    Area = ½ × 240 = 120 cm²

  4. Final Answer:

    Rhombus का area = 120 cm²

  5. Quick Check:

    Diagonals right angle पर bisect होती हैं; product of diagonals = 240 → half = 120 ✅

Quick Variations

1. Parallelogram का base या height निकालें जब area और एक dimension दिया हो।

2. Rhombus की diagonals को Pythagoras theorem से निकालें।

3. Trapezium वाले formula से missing side या height निकालें।

4. Combined figure problems (जैसे parallelogram + triangle)।

Trick to Always Use

  • Step 1 → पहले shape पहचानें (parallelogram, rhombus या trapezium)।
  • Step 2 → उस shape से जुड़ा formula लिखें।
  • Step 3 → दिए गए values को systematically substitute करें - diagonals या trapezium वाले formulas में “½” कभी न भूलें।

Summary

Summary

Quadrilaterals (Parallelogram, Rhombus, Trapezium) के लिए:

  • पहले figure का प्रकार पहचानें - formulas अलग-अलग होते हैं।
  • Parallelogram: base × height; Rhombus: ½ × d₁ × d₂; Trapezium: ½ × (a + b) × h.
  • Calculation के बाद units (cm², m²) check करें।
  • Result को वापस substitute करके या approximate geometric shape से compare करके verify करें।

Practice

(1/5)
1. Find the area of a parallelogram with base 12 cm and height 8 cm.
easy
A. 96 cm²
B. 100 cm²
C. 84 cm²
D. 90 cm²

Solution

  1. Step 1: Recall the formula for area of parallelogram.

    Area = base × height.

  2. Step 2: Substitute values.

    Area = 12 × 8 = 96 cm².

  3. Final Answer:

    Area = 96 cm² → Option A.

  4. Quick Check:

    12 × 8 = 96 ✅
Hint: Multiply base by height directly.
Common Mistakes: Using ½ × base × height (that’s for triangles).
2. The diagonals of a rhombus are 16 cm and 12 cm. Find its area.
easy
A. 96 cm²
B. 90 cm²
C. 100 cm²
D. 80 cm²

Solution

  1. Step 1: Use formula for area of rhombus.

    Area = ½ × d₁ × d₂.

  2. Step 2: Substitute values.

    Area = ½ × 16 × 12 = ½ × 192 = 96 cm².

  3. Final Answer:

    Area = 96 cm² → Option A.

  4. Quick Check:

    Half of 192 = 96 ✅
Hint: Multiply diagonals and take half.
Common Mistakes: Forgetting the ½ in the formula.
3. A trapezium has parallel sides of 10 cm and 6 cm, and height 4 cm. Find its area.
easy
A. 30 cm²
B. 32 cm²
C. 36 cm²
D. 34 cm²

Solution

  1. Step 1: Use formula for trapezium area.

    Area = ½ × (sum of parallel sides) × height.

  2. Step 2: Substitute values.

    Area = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 8 × 4 = 32 cm².

  3. Final Answer:

    Area = 32 cm² → Option B.

  4. Quick Check:

    (10 + 6)/2 = 8 × 4 = 32 ✅
Hint: Take average of parallel sides and multiply by height.
Common Mistakes: Adding height to sides instead of multiplying.
4. If the area of a parallelogram is 150 cm² and its base is 15 cm, find its height.
medium
A. 8 cm
B. 12 cm
C. 14 cm
D. 10 cm

Solution

  1. Step 1: Recall area formula.

    Area = base × height.

  2. Step 2: Substitute and rearrange.

    150 = 15 × height → height = 150 ÷ 15 = 10 cm.

  3. Final Answer:

    Height = 10 cm → Option D.

  4. Quick Check:

    15 × 10 = 150 ✅
Hint: Divide area by base to find height.
Common Mistakes: Multiplying instead of dividing.
5. In a rhombus, each side is 13 cm and one diagonal is 10 cm. Find the other diagonal.
medium
A. 20 cm
B. 22 cm
C. 24 cm
D. 18 cm

Solution

  1. Step 1: Use property of rhombus diagonals.

    Diagonals bisect each other at right angles.

  2. Step 2: Apply Pythagoras theorem.

    (d₁/2)² + (d₂/2)² = side² → (10/2)² + (d₂/2)² = 13².

  3. Step 3: Simplify.

    25 + (d₂/2)² = 169 → (d₂/2)² = 144 → d₂/2 = 12 → d₂ = 24 cm.

  4. Final Answer:

    Other diagonal = 24 cm → Option C.

  5. Quick Check:

    5² + 12² = 13² ✅
Hint: Use diagonals’ half-lengths with Pythagoras theorem.
Common Mistakes: Forgetting to divide diagonals by 2 before applying the theorem.

Mock Test

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