Class 9th Mathematics Triangles Case Studies

Case Study 1: School Playground Construction

A school was constructing a triangular flower garden in its playground. The sides of the garden were measured as 7 m, 24 m, and 25 m. The principal asked students to check if this triangle was a right-angled triangle. One student applied the Pythagoras theorem and verified the condition of right triangles.

Questions:

1. Which theorem is applicable here?
   (a) Mid-point theorem (b) Pythagoras theorem (c) Angle sum property (d) None
   Answer: (b)
2. Check if 7² + 24² = 25².
   (a) Yes (b) No (c) Sometimes (d) None
   Answer: (a)
3. If true, then which angle is 90°?
   (a) Opposite to 7 (b) Opposite to 24 (c) Opposite to 25 (d) None
   Answer: (c)
4. The largest side of a right triangle is called:
   (a) base (b) height (c) hypotenuse (d) none
   Answer: (c)
5. Here, hypotenuse = ?
   (a) 7 (b) 24 (c) 25 (d) none
   Answer: (c)

Case Study 2: Flagpole Shadow

A 15 m high flagpole casts a shadow of 8 m. A student wanted to find the distance between the top of the pole and the end of the shadow. He used the concept of right triangles.

Questions:

1. Triangle formed is:
   (a) scalene (b) right triangle (c) isosceles (d) none
   Answer: (b)
2. Distance between top and end of shadow =
   (a) √(15²+8²) (b) 23 (c) √289 (d) all of these
   Answer: (d)
3. This length is called:
   (a) base (b) perpendicular (c) hypotenuse (d) height
   Answer: (c)
4. Length = ?
   (a) 21 (b) 23 (c) 25 (d) 27
   Answer: (b)
5. Which theorem is applied?
   (a) Similarity theorem (b) Pythagoras theorem (c) Exterior angle theorem (d) None
   Answer: (b)

Case Study 3: Bridge Design

Engineers designed a triangular support for a bridge. They used two equal sides of 12 m each and a base of 10 m. This created an isosceles triangle.

Questions:

1. Triangle formed is:
   (a) scalene (b) isosceles (c) equilateral (d) right
   Answer: (b)
2. Angles opposite equal sides are:
   (a) unequal (b) equal (c) complementary (d) supplementary
   Answer: (b)
3. Base angles are called:
   (a) equal angles (b) vertex angles (c) interior angles (d) none
   Answer: (a)
4. Which property is used?
   (a) ASA (b) Angles opposite equal sides are equal (c) RHS (d) none
   Answer: (b)
5. Number of equal sides here = ?
   (a) 1 (b) 2 (c) 3 (d) none
   Answer: (b)

Case Study 4: Traffic Signal Triangle

A traffic signal board is triangular in shape with all sides equal to 6 m. The students discussed whether it is an equilateral triangle and calculated each angle.

Questions:

1. All angles in equilateral triangle are:
   (a) 45° (b) 60° (c) 90° (d) 120°
   Answer: (b)
2. Triangle with 3 equal sides =
   (a) scalene (b) isosceles (c) equilateral (d) none
   Answer: (c)
3. Each angle = ?
   (a) 50° (b) 60° (c) 70° (d) 80°
   Answer: (b)
4. Sum of all angles =
   (a) 120° (b) 180° (c) 240° (d) none
   Answer: (b)
5. Property used:
   (a) angle sum property (b) Pythagoras (c) similarity (d) none
   Answer: (a)

Case Study 5: Farmer’s Field

A farmer fenced his triangular field with sides 5 m, 12 m, and 13 m. He checked whether the field is a right triangle.

Questions:

1. Which side is longest?
   (a) 5 (b) 12 (c) 13 (d) none
   Answer: (c)
2. Square of longest side = ?
   (a) 169 (b) 144 (c) 25 (d) none
   Answer: (a)
3. Sum of squares of other sides = ?
   (a) 144 (b) 169 (c) 169 (d) none
   Answer: (c)
4. Hence triangle is:
   (a) right (b) scalene (c) isosceles (d) none
   Answer: (a)
5. Property used = ?
   (a) Pythagoras theorem (b) Exterior angle theorem (c) congruency (d) none
   Answer: (a)

Case Study 6: Tower and Ladder

A ladder is placed against a wall. Ladder = 17 m, wall height = 15 m. Find distance of ladder’s foot from wall.

Questions:

1. Triangle type =
   (a) right (b) equilateral (c) scalene (d) none
   Answer: (a)
2. Base = ?
   (a) √(17²−15²) (b) 8 (c) √64 (d) all of these
   Answer: (d)
3. Distance = ?
   (a) 7 (b) 8 (c) 9 (d) 10
   Answer: (b)
4. Hypotenuse here =
   (a) 15 (b) 17 (c) 8 (d) none
   Answer: (b)
5. Theorem applied =
   (a) Pythagoras (b) similarity (c) congruency (d) none
   Answer: (a)

Case Study 7: Congruent Triangles in Architecture

An architect designed two windows of triangular shape, each with sides 5 m, 6 m, and 7 m. A student claimed both are congruent.

Questions:

1. Which rule applies?
   (a) SSS (b) SAS (c) ASA (d) RHS
   Answer: (a)
2. Triangles with equal sides are:
   (a) congruent (b) similar (c) unequal (d) none
   Answer: (a)
3. Congruent triangles have:
   (a) equal areas (b) equal perimeters (c) equal sides/angles (d) all of these
   Answer: (d)
4. In congruence, corresponding parts are:
   (a) unequal (b) equal (c) different (d) none
   Answer: (b)
5. Property is known as:
   (a) CPCT (b) RHS (c) ASA (d) none
   Answer: (a)

Case Study 8: Road Triangle

Three roads form a triangle with angles 40°, 60°, and 80°.

Questions:

1. Type of triangle =
   (a) scalene (b) isosceles (c) equilateral (d) right
   Answer: (a)
2. Largest angle = ?
   (a) 40° (b) 60° (c) 80° (d) none
   Answer: (c)
3. Side opposite largest angle =
   (a) smallest (b) largest (c) medium (d) none
   Answer: (b)
4. Property used:
   (a) greater angle → greater side (b) smaller angle → smaller side (c) both (d) none
   Answer: (c)
5. Sum of angles = ?
   (a) 100° (b) 120° (c) 180° (d) 200°
   Answer: (c)

Case Study 9: Garden Path

A triangular path has sides 8 m, 15 m, 17 m. Students checked if it is a right triangle.

Questions:

1. Hypotenuse = ?
   (a) 8 (b) 15 (c) 17 (d) none
   Answer: (c)
2. Check: 8²+15²=?
   (a) 289 (b) 225 (c) 289 (d) none
   Answer: (c)
3. 17²=?
   (a) 225 (b) 289 (c) 289 (d) none
   Answer: (b)
4. Hence triangle = ?
   (a) right (b) isosceles (c) scalene (d) none
   Answer: (a)
5. Angle opposite to 17 = ?
   (a) 90° (b) 60° (c) 120° (d) none
   Answer: (a)

Case Study 10: Roof Support

Carpenters used two equal wooden planks of 5 m each joined at the top with base 6 m. A triangular roof was formed.

Questions:

1. Triangle formed =
   (a) isosceles (b) scalene (c) right (d) equilateral
   Answer: (a)
2. Equal sides = ?
   (a) 5,5 (b) 6,6 (c) 5,6 (d) none
   Answer: (a)
3. Angles opposite equal sides are:
   (a) equal (b) unequal (c) supplementary (d) none
   Answer: (a)
4. Triangle’s type based on sides?
   (a) 2 equal (b) 3 equal (c) 0 equal (d) none
   Answer: (a)
5. Triangle congruence rule applied?
   (a) SAS (b) ASA (c) RHS (d) SSS
   Answer: (b)