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| source: ncert_exercises |
| topic: geometry |
| class_level: class_9 |
| chapter: ch10 |
| exercise: ex10.2 |
| difficulty: intermediate |
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| Class 9 | Ch10: Heron's Formula | Exercise 10.2 |
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| Application: Area of quadrilateral = sum of areas of two triangles formed by a diagonal. |
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| Q1. A park, in the shape of a quadrilateral ABCD, has ∠C=90°, AB=9 m, BC=12 m, CD=5 m, DA=8 m. How much area does it occupy? |
| Answer: Draw diagonal BD. In △BCD (right angle at C): BD=√(BC²+CD²)=√(144+25)=√169=13 m. |
| Area(△BCD)=½×12×5=30 m². In △ABD: a=AB=9, b=AD=8, c=BD=13. s=(9+8+13)/2=15. |
| Area(△ABD)=√(15×6×7×2)=√1260=6√35 m². Total=30+6√35≈30+35.5=65.5 m². ✅ |
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| Q2. A floral design on a floor is made up of 16 tiles which are triangular, sides 9 cm, 28 cm and 35 cm. Find cost of polishing tiles at 50p per cm². |
| Answer: One tile: a=9, b=28, c=35. s=(9+28+35)/2=36. Check: 9+28=37>35 ✓. |
| Area=√(36×27×8×1)=√7776=36√6 cm². 16 tiles: Area=16×36√6=576√6 cm². |
| Cost=576√6×0.5=288√6≈₹705. (Using √6≈2.449: 288×2.449≈₹705.3.) ✅ |
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| Q3. A field is in shape of a rhombus with perimeter 400 m and one diagonal 160 m. Find area. |
| Answer: Each side=400/4=100 m. Diagonal d₁=160 m → half=80 m. Other half of d₂: √(100²-80²)=√(10000-6400)=√3600=60 m → d₂=120 m. |
| Area of rhombus=½×d₁×d₂=½×160×120=9600 m². ✅ |
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| Q4. An umbrella is made by stitching 10 triangular pieces of cloth, each of dimensions 20 cm, 50 cm and 50 cm. Find area of cloth required. |
| Answer: One triangle: a=20, b=50, c=50. s=(20+50+50)/2=60. |
| Area=√(60×40×10×10)=√240000=200√6 cm². 10 triangles: Area=2000√6 cm². ✅ |
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| Q5. A kite in shape of a square with diagonal 32 cm and isosceles triangle of base 8 cm and sides 6 cm. Find total area. |
| Answer: Square diagonal=32 → Area of square=½×32×32=512 cm². Isosceles △: a=b=6, c=8. s=(6+6+8)/2=10. Area=√(10×4×4×2)=√320=8√5 cm². Total=512+8√5 cm². ✅ |
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| Q6. A triangle and parallelogram have same base (28 cm) and same area. Triangle sides: 26 cm, 28 cm, 30 cm. Find height of parallelogram. |
| Answer: Triangle: s=(26+28+30)/2=42. Area=√(42×16×14×12)=√112896=336 cm². Parallelogram: Area=base×height=28×h=336 → h=12 cm. ✅ |
