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Triangles Quiz - CBSE Class 9 Mathematics

Free practice quiz: Triangles (CBSE Class 9 Mathematics). 19 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 9 Mathematics practice quiz has 19 multiple-choice questions on Triangles, and 19 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Angle Sum and Triangle Types

  • The interior angles of every triangle add up to 180°, so the third angle is 180° minus the other two.
  • An isosceles triangle has two equal sides, and the angles opposite those sides are equal.
  • An equilateral triangle has all sides equal, so all its angles are also equal.
Angle sum property
\[\angle A + \angle B + \angle C = 180^\circ\]
Exterior angle property
\[\text{Exterior } \angle = \text{sum of the two interior opposite angles}\]
Isosceles triangle
\[AB = AC \iff \angle C = \angle B\]

Common mistake: Matching equal angles to the wrong sides. Equal angles lie opposite the equal sides, not between them.

Triangle Inequality and Side Types

  • Three lengths form a triangle only if the sum of any two is greater than the third.
  • Compare the square of the longest side with the sum of the squares of the other two sides.
  • Equal means right-angled, smaller means acute-angled, and larger means obtuse-angled.
Triangle inequality
\[a + b > c,\; b + c > a,\; c + a > b\]
Difference form
\[|a - b| < c\]
Right triangle (c longest)
\[c^2 = a^2 + b^2\]
Obtuse triangle (c longest)
\[c^2 > a^2 + b^2\]

Common mistake: Checking only one pair of sides. All pairs must satisfy the inequality, and the longest side is the critical check.

Congruence Rules: Unique Triangles

  • SSS, SAS, ASA, AAS and RHS each fix one unique triangle, up to congruence.
  • AAA fixes only the shape, not the size. SSA with a non-included angle may give two triangles.
Valid criteria
\[\text{SSS},\ \text{SAS},\ \text{ASA},\ \text{AAS},\ \text{RHS}\]
Not valid
\[\text{AAA},\ \text{SSA}\]

Common mistake: Using SAS when the angle is not between the two given sides. That case is SSA, which is not a valid rule.

Special Lines and Centres

  • A median joins a vertex to the midpoint of the opposite side. Each vertex has exactly one median.
  • Medians meet at the centroid and angle bisectors meet at the incentre, which is the centre of the incircle.
  • Perpendicular bisectors meet at the circumcentre, whose circle passes through all vertices. Altitudes meet at the orthocentre.
Centroid divides each median
\[AG : GD = 2 : 1\]
Sum of medians
\[m_a + m_b + m_c < a + b + c\]
Circumcircle
\[OA = OB = OC\]

Common mistake: Mixing up the centres. Remember: medians give the centroid, altitudes the orthocentre, and angle bisectors the incentre.

Area of a Triangle

  • Area is half of the base times the height drawn to that base.
  • The area stays the same whichever side you take as base, so a longer base means a shorter height.
  • Heron's formula gives the area from the three sides alone.
Base and height
\[\text{Area} = \frac{1}{2} \times b \times h\]
Heights are inversely proportional to bases
\[\frac{h_1}{h_2} = \frac{b_2}{b_1}\]
Semi-perimeter
\[s = \frac{a + b + c}{2}\]
Heron's formula
\[\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\]

Common mistake: Writing the ratio of heights equal to the ratio of the sides. Since the area is fixed, the ratio is inverted.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. How many medians can a triangle have?

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    Answer: C. 3

    A triangle has exactly 3 medians, one from each vertex to the midpoint of the opposite side.

  2. Which of the following does NOT define a unique triangle?

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    Answer: D. SSA

    The SSA condition does not define a unique triangle due to the possibility of the ambiguous case; it may produce two different triangles, one triangle, or no triangle at all.

  3. If a triangle has interior angles measuring 50 degrees and 60 degrees, what is the measure of the third angle?

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    Answer: C. 70 degrees

    The sum of interior angles of a triangle is 180 degrees. Therefore, if two angles are 50 degrees and 60 degrees, the third angle must be 70 degrees (180 - 50 - 60 = 70).

  4. The orthocenter of a triangle is located at the intersection of which lines?

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    Answer: B. Altitudes

    The orthocenter of a triangle is the point where all three altitudes intersect.

  5. In any triangle, what is the sum of the median lengths related to?

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    Answer: B. It is related to 3/4 of the sum of the side lengths

    The sum of the medians of a triangle is related to approximately 3/4 of the sum of its sides, but the exact relationship varies based on the triangle's dimensions.

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. If a triangle has sides 3 cm, 4 cm, and 5 cm, what type of triangle is it?
  2. Which of the following is used to find the area of a triangle?
  3. Which type of triangle has all equal sides?
  4. In triangle ABC, if angle A = 90 degrees and angle B = 45 degrees, what is angle C?
  5. Which special point in a triangle is the intersection of the medians?
  6. The point where the angle bisectors of a triangle intersect is called the:
  7. A triangle has sides of length 7 cm, 10 cm, and 5 cm. Which inequality of a triangle does it satisfy?
  8. In triangle PQR, if PQ = QR = RP = 5 cm, what type of triangle is PQR?
  9. For a triangle with areas proportional to its sides, if one side is 3 units, and another side is 5 units, what is the ratio of the triangle's heights from these two sides?
  10. Which condition among the following is necessary for a set of numbers to be the sides of a triangle?
  11. In triangle XYZ, if angle X is 30 degrees and angle Y is 90 degrees, what is the ratio of XY to XZ?
  12. What circle passes through all vertices of a triangle?
  13. In triangle ABC, if AB = AC and angle B is twice angle A, what type of triangle is ABC?
  14. In triangle DEF, DE = EF = DF and angle D is 60 degrees. What is the measure of angle E?

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