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Class 9 Inequalities in a Triangle — practice questions with answers

Geometry — Triangles · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.

1. In any triangle, the side opposite the greater angle is:
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Answer: A) the longer side
Theorem: In a triangle, the side opposite to the greater angle is longer. So a greater angle faces a longer side.
2. Two sides of a triangle are 8 cm and 5 cm. The third side must be less than:
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Answer: A) 13 cm
Third side < sum of the two sides = 8 + 5 = 13 cm. So the third side must be less than 13 cm.
3. The perimeter of a triangle is always _____ the sum of its three medians? (choose the correct comparison for two sides vs third)
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Answer: A) sum of two sides > third side
By the triangle inequality, the sum of any two sides exceeds the third side. This is the basic side-length inequality for triangles.
4. A triangle has two sides measuring 10 cm and 10 cm.
For how many integer values of the third side does a valid triangle exist?
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Answer: A) 19
Third side x satisfies 10-10 < x < 10+10, i.e. 0 < x < 20. Integer values 1 to 19 inclusive = 19 values.
5. In triangle ABC, point P is any point inside the triangle.
Using the result that the sum of distances from an interior point to the vertices is less than the perimeter, which inequality is valid?
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Answer: A) PA + PB + PC < AB + BC + CA
For an interior point P, extend BP to meet AC; using triangle inequalities one shows PA + PB + PC is less than the full perimeter AB + BC + CA. (It also exceeds half the perimeter, but the upper bound asked is the full perimeter.)
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