Case Study 1: Garden Triangular Plot
A farmer owns a triangular garden plot with sides 13 m, 14 m, and 15 m. He wants to calculate its area to know how much grass seed to buy. He decides to apply Heron’s formula.
Questions:
- Semi-perimeter, ss = ?
(a) 21 (b) 20 (c) 22 (d) 19
Answer: (a) - Expression under root = ?
(a) 21×8×7×6 (b) 21×7×6×5 (c) 21×8×6×7 (d) none
Answer: (a) - Area = ?
(a) 84 m² (b) 90 m² (c) 100 m² (d) 120 m²
Answer: (a) - Formula used = ?
(a) Pythagoras theorem (b) Heron’s formula (c) Area = ½ × base × height (d) none
Answer: (b) - Shape of garden = ?
(a) quadrilateral (b) triangle (c) circle (d) trapezium
Answer: (b)
Case Study 2: Playground Division
A school divides a quadrilateral playground into two triangles along a diagonal. Triangles are ∆ABC (sides 7, 8, 9 m) and ∆ADC (sides 6, 8, 10 m).
Questions:
- To find total area of quadrilateral, we:
(a) add areas of triangles (b) multiply areas (c) subtract (d) none
Answer: (a) - For ∆ABC, semi-perimeter = ?
(a) 12 (b) 13 (c) 14 (d) 15
Answer: (a) - For ∆ADC, semi-perimeter = ?
(a) 12 (b) 13 (c) 14 (d) 15
Answer: (b) - Formula applied in each triangle = ?
(a) Heron’s formula (b) Pythagoras (c) mid-point theorem (d) none
Answer: (a) - This method is useful for:
(a) irregular plots (b) only squares (c) only rectangles (d) none
Answer: (a)
Case Study 3: Flag Decoration
During Independence Day, a triangular flag with sides 9 m, 12 m, and 15 m was made.
Questions:
- Semi-perimeter = ?
(a) 16 (b) 18 (c) 20 (d) 15
Answer: (b) - This triangle is also a:
(a) right-angled triangle (b) isosceles (c) equilateral (d) none
Answer: (a) - Area using Heron’s formula = ?
(a) 54 (b) 72 (c) 90 (d) 108
Answer: (c) - Area using ½ × base × height = ?
(a) 90 (b) 100 (c) 120 (d) 108
Answer: (a) - Which method is easier here?
(a) Heron’s formula (b) ½ × base × height (c) both give same answer (d) none
Answer: (c)
Case Study 4: Triangular Park
A triangular park has sides 7 m, 24 m, and 25 m. The municipal corporation wants to cover it with grass.
Questions:
- Type of triangle = ?
(a) right-angled (b) equilateral (c) isosceles (d) scalene
Answer: (a) - Semi-perimeter = ?
(a) 28 (b) 27 (c) 26 (d) 25
Answer: (b) - Area using Heron’s formula = ?
(a) 84 (b) 72 (c) 60 (d) 100
Answer: (a) - Area using ½ × base × height = ?
(a) 84 (b) 72 (c) 60 (d) 100
Answer: (a) - Which property verified?
(a) Heron’s formula works for right triangles too (b) Pythagoras only (c) none
Answer: (a)
Case Study 5: Hall Roof
The triangular roof of a hall has sides 8 m, 15 m, and 17 m. The architect wants to paint it.
Questions:
- Semi-perimeter = ?
(a) 19 (b) 20 (c) 21 (d) 22
Answer: (b) - Type of triangle = ?
(a) right-angled (b) isosceles (c) equilateral (d) none
Answer: (a) - Area = ?
(a) 60 (b) 65 (c) 70 (d) 72
Answer: (a) - Formula used = ?
(a) Heron’s formula (b) ½ × base × height (c) both (d) none
Answer: (c) - Roof shape is:
(a) triangular (b) quadrilateral (c) polygon (d) none
Answer: (a)
Case Study 6: Farming Land
A triangular field has sides 30 m, 40 m, and 50 m. A farmer wants to calculate area for crop planning.
Questions:
- Semi-perimeter = ?
(a) 60 (b) 50 (c) 70 (d) 80
Answer: (a) - This triangle is:
(a) right-angled (b) scalene (c) equilateral (d) none
Answer: (a) - Area = ?
(a) 600 (b) 650 (c) 700 (d) 750
Answer: (c) - Area using ½ × base × height = ?
(a) 600 (b) 650 (c) 700 (d) 750
Answer: (c) - Heron’s formula is useful for:
(a) all triangles (b) only right-angled triangles (c) only equilateral triangles (d) none
Answer: (a)
Case Study 7: Playground Boundary
A triangular playground has sides 10 m, 10 m, and 12 m. Students want to calculate area to install synthetic grass.
Questions:
- Triangle type = ?
(a) isosceles (b) scalene (c) equilateral (d) none
Answer: (a) - Semi-perimeter = ?
(a) 15 (b) 16 (c) 17 (d) 18
Answer: (b) - Area = ?
(a) 40 (b) 48 (c) 60 (d) 72
Answer: (b) - Heron’s formula works for:
(a) scalene only (b) isosceles only (c) all triangles (d) none
Answer: (c) - Which side is repeated?
(a) 10 (b) 12 (c) 8 (d) none
Answer: (a)
Case Study 8: Road Triangle
Three roads form a triangle with sides 9 km, 40 km, and 41 km.
Questions:
- Semi-perimeter = ?
(a) 40 (b) 44 (c) 45 (d) 46
Answer: (b) - Type of triangle = ?
(a) right-angled (b) isosceles (c) equilateral (d) none
Answer: (a) - Area = ?
(a) 180 (b) 160 (c) 150 (d) 200
Answer: (a) - This problem shows application in:
(a) construction (b) road planning (c) agriculture (d) all
Answer: (d) - Formula used = ?
(a) Heron’s formula (b) ½ × base × height (c) both (d) none
Answer: (c)
Case Study 9: Cinema Screen
A triangular cinema screen has sides 7 m, 24 m, 25 m.
Questions:
- Type of triangle = ?
(a) right-angled (b) scalene (c) isosceles (d) none
Answer: (a) - Semi-perimeter = ?
(a) 28 (b) 27 (c) 26 (d) none
Answer: (b) - Area = ?
(a) 70 (b) 80 (c) 84 (d) 90
Answer: (c) - Method used = ?
(a) Heron’s formula (b) ½ × base × height (c) both (d) none
Answer: (c) - Heron’s formula helps where:
(a) height not known (b) base not known (c) only sides known (d) all
Answer: (d)
Case Study 10: Survey Land
A surveyor measures a triangular land with sides 11 m, 13 m, and 20 m.
Questions:
- Semi-perimeter = ?
(a) 20 (b) 21 (c) 22 (d) 23
Answer: (b) - Expression inside root = ?
(a) 21×10×8×2 (b) 21×11×7×3 (c) 21×9×8×4 (d) none
Answer: (a) - Area = ?
(a) 36√10 (b) 40√10 (c) 42√10 (d) none
Answer: (a) - This application is useful for:
(a) surveyors (b) architects (c) farmers (d) all
Answer: (d) - Formula always applicable for:
(a) only equilateral (b) only isosceles (c) all triangles (d) none
Answer: (c)