Triangles are fundamental shapes in geometry, and they play a crucial role in Class 10 Mathematics. Understanding the properties of triangles, such as congruence and similarity, is essential for solving geometric problems. The following MCQs will help students assess their grasp on the topic and prepare thoroughly for exams.
Chapter: Triangles
1. Two triangles are said to be congruent if:
- A) Their corresponding angles are equal and corresponding sides are proportional
- B) Their corresponding angles are equal and corresponding sides are equal
- C) Their corresponding sides are equal
- D) Their corresponding areas are equal
Answer: B) Their corresponding angles are equal and corresponding sides are equal
Congruent triangles are identical in shape and size. Both corresponding sides and angles are equal.
2. Which of the following is a criterion for the congruence of triangles?
- A) SAS
- B) SSS
- C) ASA
- D) All of the above
Answer: D) All of the above
The congruence criteria for triangles include SAS (Side-Angle-Side), SSS (Side-Side-Side), and ASA (Angle-Side-Angle).
3. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. This is a property of:
- A) Congruent triangles
- B) Similar triangles
- C) Right triangles
- D) Scalene triangles
Answer: B) Similar triangles
For similar triangles, the ratio of their areas is the square of the ratio of their corresponding sides.
4. If two triangles are congruent, their:
- A) Corresponding angles are equal
- B) Corresponding sides are equal
- C) Both A and B
- D) None of the above
Answer: C) Both A and B
In congruent triangles, both corresponding angles and corresponding sides are equal.
5. The sum of the angles of a triangle is:
- A) 180°
- B) 90°
- C) 360°
- D) 270°
Answer: A) 180°
In every triangle, the sum of the three interior angles is always 180°.
6. If in triangle ABC, AB = AC, then triangle ABC is:
- A) Scalene
- B) Isosceles
- C) Equilateral
- D) Right-angled
Answer: B) Isosceles
An isosceles triangle has two equal sides. In this case, AB = AC, making the triangle isosceles.
7. In an equilateral triangle, each angle measures:
- A) 90°
- B) 60°
- C) 45°
- D) 120°
Answer: B) 60°
In an equilateral triangle, all three angles are equal, and each measures 60°.
8. The Pythagorean theorem holds true for:
- A) Right-angled triangles
- B) Acute-angled triangles
- C) Obtuse-angled triangles
- D) All triangles
Answer: A) Right-angled triangles
The Pythagorean theorem applies only to right-angled triangles, where c2=a2+b2c^2 = a^2 + b^2c2=a2+b2 for sides a, b, and hypotenuse c.
9. In a triangle, if two sides are equal, then the angles opposite those sides are:
- A) Equal
- B) Different
- C) One is 90°
- D) None of the above
Answer: A) Equal
In an isosceles triangle, the angles opposite the equal sides are equal.
10. The area of an equilateral triangle with side length ‘a’ is:
- A) a22\frac{a^2}{2}2a2
- B) 34a2\frac{\sqrt{3}}{4} a^243a2
- C) a2a^2a2
- D) a24\frac{a^2}{4}4a2
Answer: B) 34a2\frac{\sqrt{3}}{4} a^243a2
The area of an equilateral triangle is given by the formula A=34a2A = \frac{\sqrt{3}}{4} a^2A=43a2, where ‘a’ is the side length.
11. If the ratio of the sides of two similar triangles is 3:4, then the ratio of their areas is:
- A) 9:16
- B) 16:9
- C) 12:16
- D) 3:4
Answer: A) 9:16
For similar triangles, the ratio of their areas is the square of the ratio of their corresponding sides. So, (32):(42)=9:16(3^2):(4^2) = 9:16(32):(42)=9:16.
12. In an isosceles triangle, the altitude from the vertex bisects:
- A) The base and the angle at the vertex
- B) The base only
- C) The angles at the base
- D) None of the above
Answer: A) The base and the angle at the vertex
In an isosceles triangle, the altitude from the vertex bisects both the base and the angle at the vertex.
13. In a triangle ABC, if ∠A=90∘\angle A = 90^\circ∠A=90∘, then triangle ABC is:
- A) Right-angled triangle
- B) Obtuse triangle
- C) Acute triangle
- D) Scalene triangle
Answer: A) Right-angled triangle
A triangle with one angle measuring 90° is a right-angled triangle.
14. The longest side of a right triangle is called the:
- A) Hypotenuse
- B) Adjacent
- C) Opposite
- D) Altitude
Answer: A) Hypotenuse
In a right triangle, the longest side is always opposite the right angle, and it is called the hypotenuse.
15. The midpoint theorem states that the line joining the midpoints of two sides of a triangle is:
- A) Parallel to the third side and half of it
- B) Perpendicular to the third side
- C) Equal to the third side
- D) None of the above
Answer: A) Parallel to the third side and half of it
The midpoint theorem states that the line joining the midpoints of two sides of a triangle is parallel to the third side and is half its length.
16. In a triangle, if one side is longer than another, the angle opposite the longer side is:
- A) Larger
- B) Smaller
- C) Equal
- D) Undefined
Answer: A) Larger
In any triangle, the larger angle is opposite the longer side.
17. In a right-angled triangle, the two sides forming the right angle are called:
- A) Hypotenuse
- B) Base and height
- C) Legs
- D) Adjacent sides
Answer: C) Legs
In a right-angled triangle, the sides that form the right angle are known as the legs.
18. If the area of a triangle is 30 square units and the base is 6 units, the height is:
- A) 5
- B) 6
- C) 10
- D) 15
Answer: A) 5
The area of a triangle is given by Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}Area=21×base×height. Substituting the values: 30=12×6×height30 = \frac{1}{2} \times 6 \times \text{height}30=21×6×height, the height is 5 units.
19. If two triangles are similar, the ratio of their corresponding sides is:
- A) Always 1
- B) Greater than 1
- C) Equal to 1
- D) Constant
Answer: D) Constant
In similar triangles, the ratio of corresponding sides is constant.
20. The sum of any two sides of a triangle is:
- A) Greater than the third side
- B) Equal to the third side
- C) Less than the third side
- D) None of the above
Answer: A) Greater than the third side
The sum of any two sides of a triangle is always greater than the length of the third side.
21. If in triangle ABC, AB=ACAB = ACAB=AC, then triangle ABC is:
- A) Scalene
- B) Equilateral
- C) Isosceles
- D) Right-angled
Answer: C) Isosceles
An isosceles triangle has two equal sides, and here AB = AC.
22. In a right-angled triangle, if the lengths of the two perpendicular sides are 3 and 4, the length of the hypotenuse is:
- A) 4
- B) 5
- C) 6
- D) 7
Answer: B) 5
By the Pythagorean theorem, c2=a2+b2c^2 = a^2 + b^2c2=a2+b2, so c2=32+42=9+16=25c^2 = 3^2 + 4^2 = 9 + 16 = 25c2=32+42=9+16=25, so c=5c = 5c=5.
23. If the area of an equilateral triangle is 81√3 square units, the side length is:
- A) 9
- B) 12
- C) 18
- D) 6
Answer: A) 9
For an equilateral triangle, Area=34a2\text{Area} = \frac{\sqrt{3}}{4} a^2Area=43a2. Solving the equation for side length a=9a = 9a=9.
24. In a triangle, if one angle is 90°, then it is:
- A) Acute triangle
- B) Right-angled triangle
- C) Obtuse triangle
- D) Isosceles triangle
Answer: B) Right-angled triangle
A triangle with one angle of 90° is classified as a right-angled triangle.
25. The number of sides of a triangle is:
- A) 2
- B) 3
- C) 4
- D) 6
Answer: B) 3
A triangle has three sides.
26. In a triangle, the sum of the two sides is always:
- A) Less than the third side
- B) Greater than the third side
- C) Equal to the third side
- D) Zero
Answer: B) Greater than the third side
The sum of any two sides of a triangle is always greater than the length of the third side.
27. If two triangles are congruent, then their:
- A) Corresponding sides and angles are not equal
- B) Corresponding sides and angles are equal
- C) Corresponding sides are not equal but angles are equal
- D) None of the above
Answer: B) Corresponding sides and angles are equal
Congruent triangles have exactly the same size and shape, meaning all corresponding sides and angles are equal.
28. Which of the following is not a similarity criterion for triangles?
- A) SAS (Side-Angle-Side)
- B) AAA (Angle-Angle-Angle)
- C) SSS (Side-Side-Side)
- D) SSA (Side-Side-Angle)
Answer: D) SSA (Side-Side-Angle)
SSA is not a similarity criterion. The criteria for similarity are AAA, SAS, and SSS.
29. If two triangles are similar, then the ratio of their corresponding heights is:
- A) The same as the ratio of their corresponding sides
- B) Half of the ratio of their corresponding sides
- C) The same as the ratio of their areas
- D) None of the above
Answer: A) The same as the ratio of their corresponding sides
In similar triangles, the ratio of corresponding heights is the same as the ratio of corresponding sides.
30. In an equilateral triangle, all sides are:
- A) Equal
- B) Different
- C) Perpendicular to each other
- D) Of integer values
Answer: A) Equal
In an equilateral triangle, all three sides are of equal length.
31. If in a triangle ABC, the sides AB = AC, then which of the following is true?
- A) Angle B = Angle C
- B) Angle A = Angle B
- C) Angle B = Angle A
- D) None of the above
Answer: A) Angle B = Angle C
In an isosceles triangle, the angles opposite the equal sides are equal. So, Angle B = Angle C.
32. The area of a right triangle is equal to:
- A) 12×base×height\frac{1}{2} \times \text{base} \times \text{height}21×base×height
- B) base×height\text{base} \times \text{height}base×height
- C) 14×base×height\frac{1}{4} \times \text{base} \times \text{height}41×base×height
- D) base2+height2\text{base}^2 + \text{height}^2base2+height2
Answer: A) 12×base×height\frac{1}{2} \times \text{base} \times \text{height}21×base×height
The area of any triangle, including a right triangle, is calculated as 12×base×height\frac{1}{2} \times \text{base} \times \text{height}21×base×height.
33. If the corresponding sides of two triangles are in the ratio 5:7, then the ratio of their areas is:
- A) 25:49
- B) 5:7
- C) 7:5
- D) 49:25
Answer: A) 25:49
The ratio of areas of two similar triangles is the square of the ratio of their corresponding sides. So, 52:72=25:495^2 : 7^2 = 25 : 4952:72=25:49.
34. In a triangle, the line joining the midpoints of two sides is parallel to:
- A) The third side
- B) The longest side
- C) The perpendicular from the vertex
- D) None of the above
Answer: A) The third side
The line joining the midpoints of two sides of a triangle is parallel to the third side and is half its length (Midpoint Theorem).
35. If in triangle ABC, ∠A=90∘\angle A = 90^\circ∠A=90∘, then triangle ABC is:
- A) Scalene
- B) Right-angled triangle
- C) Obtuse triangle
- D) Acute triangle
Answer: B) Right-angled triangle
A triangle with one angle equal to 90° is a right-angled triangle.
36. In triangle ABC, if AB=ACAB = ACAB=AC, then triangle ABC is:
- A) Right-angled
- B) Isosceles
- C) Scalene
- D) Equilateral
Answer: B) Isosceles
In an isosceles triangle, two sides are equal. Here, AB=ACAB = ACAB=AC, so triangle ABC is isosceles.
37. The hypotenuse of a right-angled triangle is:
- A) The side opposite the right angle
- B) The longest side
- C) Equal to the sum of the other two sides
- D) The shortest side
Answer: B) The longest side
The hypotenuse is the longest side in a right-angled triangle and is opposite the right angle.
38. The base of an equilateral triangle is 8 cm. What is its area?
- A) 32 cm232 \, \text{cm}^232cm2
- B) 16 cm216 \, \text{cm}^216cm2
- C) 163 cm216\sqrt{3} \, \text{cm}^2163cm2
- D) 323 cm232\sqrt{3} \, \text{cm}^2323cm2
Answer: C) 163 cm216\sqrt{3} \, \text{cm}^2163cm2
The area of an equilateral triangle is 34a2\frac{\sqrt{3}}{4} a^243a2, where aaa is the side length. Here, a=8a = 8a=8, so the area is 34×82=163 cm2\frac{\sqrt{3}}{4} \times 8^2 = 16\sqrt{3} \, \text{cm}^243×82=163cm2.
39. The angles of a triangle are in the ratio 2:3:4. The smallest angle is:
- A) 40°
- B) 60°
- C) 80°
- D) 90°
Answer: A) 40°
The sum of the angles of any triangle is 180°. Let the angles be 2x,3x,4x2x, 3x, 4x2x,3x,4x. So, 2x+3x+4x=180°2x + 3x + 4x = 180°2x+3x+4x=180°. Solving for x=20°x = 20°x=20°, the smallest angle is 2x=40°2x = 40°2x=40°.
40. In triangle ABC, if AB = 5 cm, BC = 12 cm, and AC = 13 cm, then triangle ABC is:
- A) Scalene
- B) Right-angled
- C) Equilateral
- D) Isosceles
Answer: B) Right-angled
This is a Pythagorean triplet (5, 12, 13), so triangle ABC is a right-angled triangle.
41. If a triangle has angles of 35°, 65°, and 80°, it is a:
- A) Right triangle
- B) Acute triangle
- C) Obtuse triangle
- D) Scalene triangle
Answer: B) Acute triangle
All the angles in the triangle are less than 90°, so it is an acute triangle.
42. In an isosceles triangle, the angle between the two equal sides is 40°. The other two angles are:
- A) 70°
- B) 80°
- C) 60°
- D) 50°
Answer: B) 80°
In an isosceles triangle, the angles opposite the equal sides are equal. The sum of angles in a triangle is 180°, so the other two angles must be 180°−40°2=80°\frac{180° – 40°}{2} = 80°2180°−40°=80° each.
43. The sum of the interior angles of a triangle is always:
- A) 360°
- B) 180°
- C) 90°
- D) None of the above
Answer: B) 180°
The sum of the interior angles of any triangle is always 180°.
44. In triangle ABC, ∠A=60∘,∠B=90∘\angle A = 60^\circ, \angle B = 90^\circ∠A=60∘,∠B=90∘, and ∠C=30∘\angle C = 30^\circ∠C=30∘, the triangle is:
- A) Acute-angled
- B) Right-angled
- C) Obtuse-angled
- D) Isosceles
Answer: B) Right-angled
This triangle has a 90° angle, making it a right-angled triangle.
45. If in triangle ABC, the median from vertex A is drawn, it will divide the triangle into two triangles that have:
- A) Equal areas
- B) Equal perimeters
- C) Equal angles
- D) None of the above
Answer: A) Equal areas
The median of a triangle divides it into two triangles of equal area.
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