SSC Triangle Questions with Solutions (Properties, Centres and Similarity)

Triangles are the heart of SSC geometry, because almost every geometry chapter uses them. Questions come from angles, Pythagoras theorem, similar triangles, the midpoint and angle bisector theorems and the four triangle centres. This practice paper gives you 21 triangle questions with answers and short solutions.

Practice Paper at a Glance

Types of Triangles and Basic Properties

A triangle has three sides and three angles whose sum is always 180 degrees.

• By sides: equilateral (all sides equal), isosceles (two equal) and scalene (all different).
• By angles: acute, right (one angle 90 degrees) and obtuse (one angle more than 90 degrees).
• The sum of any two sides is greater than the third side.
• The exterior angle equals the sum of the two opposite interior angles.

Formulas and Shortcut Tricks

• Pythagoras: hypotenuse^2 = base^2 + height^2 (triples: 3-4-5, 5-12-13, 8-15-17).
• Area = 1/2 x base x height; Heron: square root of s(s-a)(s-b)(s-c); equilateral: (root 3 / 4) x a^2.
• Midpoint theorem: the line joining the midpoints of two sides is parallel to the third side and half of it.
• Basic proportionality theorem: a line parallel to one side divides the other two sides in the same ratio.
• Angle bisector theorem: BD / DC = AB / AC.
• Similar triangles: ratio of areas = square of the ratio of sides.
• Centres: incentre angle BIC = 90 + A/2, circumcentre angle BOC = 2A, orthocentre angle BHC = 180 - A. The centroid divides a median 2 : 1.
• Inradius = area / s. Circumradius of a right triangle = hypotenuse / 2.

Set 1: Basic Triangle Questions (Q1–Q7)

Q1. The angles of a triangle are in the ratio 1 : 2 : 3. Find the largest angle.
Answer: 90 degrees
Solution: Parts = 6 and each part = 30 degrees, so the largest = 90 degrees.

Q2. The exterior angle of a triangle is 120 degrees and one opposite interior angle is 50 degrees. Find the other.
Answer: 70 degrees
Trick: 120 - 50 = 70 degrees.

Q3. The legs of a right triangle are 9 cm and 12 cm. Find the hypotenuse.
Answer: 15 cm
Solution: It is the 3-4-5 triple multiplied by 3.

Q4. Find the height of an equilateral triangle of side 6 cm.
Answer: 3 root 3 cm (about 5.2 cm)
Solution: Height = (root 3 / 2) x 6 = 3 root 3 cm.

Q5. Find the area of a triangle with sides 5 cm, 12 cm and 13 cm.
Answer: 30 sq cm
Solution: It is a right triangle, so area = 1/2 x 5 x 12 = 30 sq cm.

Q6. The equal sides of an isosceles triangle are 13 cm and its base is 10 cm. Find its height.
Answer: 12 cm
Solution: Height = square root of (13^2 - 5^2) = 12 cm.

Q7. Can the sides 4 cm, 5 cm and 10 cm form a triangle?
Answer: No
Solution: 4 + 5 = 9, which is less than 10.

Set 2: Theorems and Similarity (Q8–Q14)

Q8. D and E are the midpoints of AB and AC in triangle ABC. If BC = 18 cm, find DE.
Answer: 9 cm
Trick: By the midpoint theorem, DE = BC / 2.

Q9. In triangle ABC, DE is parallel to BC. If AD = 4, DB = 6 and AE = 6, find EC.
Answer: 9
Solution: AD / DB = AE / EC gives 4/6 = 6/EC, so EC = 9.

Q10. In triangle ABC, AD bisects angle A. If AB = 8, AC = 12 and BC = 15, find BD.
Answer: 6
Trick: BD / DC = 8 / 12 = 2 / 3, so BD = 15 x 2/5 = 6.

Q11. The areas of two similar triangles are in the ratio 4 : 9. If a side of the smaller triangle is 6 cm, find the corresponding side of the larger one.
Answer: 9 cm
Solution: The ratio of sides = 2 : 3, so the side = 6 x 3/2 = 9 cm.

Q12. Find the length of the median to the hypotenuse of a right triangle with legs 6 cm and 8 cm.
Answer: 5 cm
Solution: The median to the hypotenuse is half of the hypotenuse (10 / 2).

Q13. The centroid divides a median of length 15 cm. Find its distance from the vertex.
Answer: 10 cm
Solution: The centroid divides the median in the ratio 2 : 1 from the vertex.

Q14. O is the incentre of triangle ABC and angle A = 50 degrees. Find angle BOC.
Answer: 115 degrees
Trick: 90 + 50/2 = 115 degrees.

Set 3: Exam-Level Questions (Q15–Q21)

Q15. O is the circumcentre of triangle ABC and angle A = 60 degrees. Find angle BOC.
Answer: 120 degrees
Solution: Angle BOC = 2 x angle A.

Q16. H is the orthocentre of triangle ABC and angle A = 70 degrees. Find angle BHC.
Answer: 110 degrees
Solution: Angle BHC = 180 - 70 = 110 degrees.

Q17. Find the inradius of a triangle with sides 13 cm, 14 cm and 15 cm.
Answer: 4 cm
Solution: Area = 84 and s = 21, so r = 84 / 21 = 4 cm.

Q18. Find the circumradius of an equilateral triangle of side 6 cm.
Answer: 2 root 3 cm
Solution: R = a / root 3 = 6 / root 3 = 2 root 3 cm.

Q19. Find the inradius of an equilateral triangle of side 6 cm.
Answer: root 3 cm
Solution: r = a / (2 root 3) = root 3 cm.

Q20. Find the area of an equilateral triangle of side 10 cm.
Answer: 25 root 3 sq cm
Solution: Area = (root 3 / 4) x 100 = 25 root 3 sq cm.

Q21. In a right triangle, the altitude to the hypotenuse divides it into 4 cm and 9 cm. Find the altitude.
Answer: 6 cm
Trick: Altitude^2 = 4 x 9 = 36, so altitude = 6 cm.

Common Mistakes in Triangle Questions

• Using the wrong formula for the angle at the incentre, circumcentre and orthocentre.
• Taking the ratio of areas of similar triangles equal to the ratio of sides.
• Forgetting to check the triangle inequality in "can a triangle be formed" questions.
• Using the median formula for the altitude.
• Not drawing a clean figure before solving.
• Mixing up the circumradius and inradius formulas.

How to Practice These 21 Questions

1. Write all triangle theorems on one page.
2. Practise 10 questions on each theorem.
3. Solve each set in 8 minutes.
4. Redraw the figures of the questions you got wrong.
5. Revise the centre formulas daily for a week.

Related Topics to Study Next

• Mensuration Questions
• Trigonometry Questions
• SSC Algebra Questions
• Fractions and Decimals Questions

Frequently Asked Questions

What is the angle sum property of a triangle?

The three angles of a triangle always add up to 180 degrees.

What is the Pythagoras theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

What is the midpoint theorem?

The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

What is the angle bisector theorem?

The bisector of an angle divides the opposite side in the ratio of the other two sides.

What is the ratio of areas of similar triangles?

It is equal to the square of the ratio of their corresponding sides.

What is the centroid?

The centroid is the point where the three medians meet. It divides each median in the ratio 2 : 1.

What are the incentre, circumcentre and orthocentre?

The incentre is the centre of the inscribed circle, the circumcentre is the centre of the circumscribed circle and the orthocentre is the point where the altitudes meet.

What is Heron formula?

Area = square root of s(s - a)(s - b)(s - c), where s is half of the perimeter.

What is the formula for the inradius of a triangle?

Inradius = area / semi-perimeter.

Which Pythagorean triples are most useful?

3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.

How many triangle questions come in SSC exams?

Several geometry questions per shift involve triangles, so it is a high-weightage topic in CGL.

How should I prepare triangles for SSC exams?

Learn the theorems and formulas, draw figures, solve topic-wise questions and revise previous year geometry questions.