Triangles — Case Study Based Questions

We need to have atleast three parameters to construct a triangle. But still there are certain limitations. The angle sum property of a triangle and the sum of any two sides of a triangle being always greater than the third side, play an important role in judging whether a triangle can be constructed or not.

1. If the lengths of two sides of a triangle are 14 cm and 18 cm, then the length of the third side can be :
   (a) 32 cm
   (b) 5 cm
   (c) 4 cm
   (d) 3 cm

2. In which of the following cases is the construction of △ABC possible?
   (a) AB = 3 cm, BC = 5 cm, AC = 8 cm
   (b) AB = 4 cm, BC = 5 cm, AC = 10 cm
   (c) AB = 5 cm, BC = 6 cm, AC = 7 cm
   (d) None of these

3. In which of the following cases is the construction of △ABC possible?
   (a) ∠A = 50°, ∠B = 60°, ∠C = 80°
   (b) ∠A = 40°, ∠B = 50°, ∠C = 60°
   (c) ∠A = 55°, ∠B = 55°, ∠C = 55°
   (d) ∠A = 35°, ∠B = 45°, ∠C = 100°

4. In which of the following cases is the construction of △ABC possible?
   (a) AB = 5 cm, BC = 6 cm, ∠C = 40°
   (b) AB = 6 cm, BC = 8 cm, ∠B = 50°
   (c) AB = 4 cm, BC = 7 cm, ∠A = 60°
   (d) None of these

Answer

1. The length of the third side must be less than the sum and greater than the difference of the other two sides.

Sum of the two given sides = 18 cm + 14 cm = 32 cm

Difference of the two given sides = 18 cm − 14 cm = 4 cm

So, the length of the third side must be greater than 4 cm and less than 32 cm. Among the given options, only 5 cm satisfies this condition.

Hence, option (b) is the correct option.

2. The construction is possible only if the sum of any two sides is greater than the third side.

⇒ For (a): 3 + 5 = 8, which is not greater than 8.

⇒ For (b): 4 + 5 = 9 < 10.

⇒ For (c): 5 + 6 = 11 > 7, and every other pair also has a sum greater than the third side.

Only the sides given in option (c) satisfy the condition.

Hence, option (c) is the correct option.

3. The construction is possible only if the sum of the three angles is 180°.

⇒ For (a): 50° + 60° + 80° = 190°.

⇒ For (b): 40° + 50° + 60° = 150°.

⇒ For (c): 55° + 55° + 55° = 165°.

⇒ For (d): 35° + 45° + 100° = 180°.

Only the angles given in option (d) add up to 180°.

Hence, option (d) is the correct option.

4. A triangle can be constructed uniquely when two sides and the angle included between them are given.

⇒ In option (b), the two sides AB and BC are given along with ∠B, which is the angle included between them. This is the case of two sides and the included angle, so the construction is possible.

⇒ In option (a), ∠C is not the angle included between AB and BC, and in option (c), ∠A is not the angle included between AB and BC, so these do not give a determinate triangle.

Hence, option (b) is the correct option.