Properties of Triangles MCQ Quiz - Objective Question with Answer for Properties of Triangles - Download Free PDF

Last updated on Jun 20, 2025

Latest Properties of Triangles MCQ Objective Questions

Properties of Triangles Question 1:

Comprehension:

Consider the following for the two (02) items that follow: Let ABC be a triangle right-angled at B and AB+AC = 3 units.

What is the maximum area of the triangle?

Answer (Detailed Solution Below)

Option 1 :

Properties of Triangles Question 1 Detailed Solution

Calculation:

Given,

Let and .

Then,

The area of the triangle is,

To maximize, differentiate w.r.t. and set to zero:

At , we get , so

∴ The maximum area is  square unit.

Hence, the correct answer is Option  1.

Properties of Triangles Question 2:

Comprehension:

Consider the following for the two (02) items that follow: Let ABC be a triangle right-angled at B and AB+AC = 3 units.

What is A equal to if the area of the triangle is maximum?

  1. π/6
  2. π/4
  3. π/3
  4. 5π/12

Answer (Detailed Solution Below)

Option 3 : π/3

Properties of Triangles Question 2 Detailed Solution

Calculation:

Given,

Let and .

Then,

The area of the triangle is,

To maximize, set up

Hence (discarding x=0, so

Therefore,

∴ .

Hence, the correct answer is Option 3.

Properties of Triangles Question 3:

Consider the following for the two (02) items that follow: Let ABC be a triangle right-angled at B and AB+AC = 3 units. What is the maximum area of the triangle?

  1.  square unit
  2.  square unit
  3.  square unit
  4.  square unit

Answer (Detailed Solution Below)

Option 1 :  square unit

Properties of Triangles Question 3 Detailed Solution

Calculation:

Given,

Let and .

Then,

The area of the triangle is,

To maximize, differentiate w.r.t. and set to zero:

At , we get , so

∴ The maximum area is  square unit.

Hence, the correct answer is Option  1.

Properties of Triangles Question 4:

Consider the following for the two (02) items that follow: Let ABC be a triangle right-angled at B and AB+AC = 3 units.

What is A equal to if the area of the triangle is maximum?

  1. π/6
  2. π/4
  3. π/3
  4. 5π/12

Answer (Detailed Solution Below)

Option 3 : π/3

Properties of Triangles Question 4 Detailed Solution

Calculation:

Given,

Let and .

Then,

The area of the triangle is,

To maximize, set up

Hence (discarding x=0, so

Therefore,

∴ .

Hence, the correct answer is Option 3.

Properties of Triangles Question 5:

Comprehension:

Consider the following for the two (02) items that follow:
In a triangle ABC, two sides BC and CA are in the ratio 2:1 and their opposite corresponding angles are in the ratio 3: 1.

Consider the following statements:

I. The triangle is right-angled.

II. One of the sides of the triangle is 3 times the other.

III. The angles A, C and B of the triangle are in AP.

Which of the statements given above is/are correct?

  1. I only
  2. II and III only
  3. I and III only
  4. I, II and III

Answer (Detailed Solution Below)

Option 3 : I and III only

Properties of Triangles Question 5 Detailed Solution

Explanation:

We are given the triangle with angles:

Step 1: Check if the sum of the angles is 180°:

This confirms that the angles satisfy the angle sum property of a triangle.

Statement I. The triangle is right-angled.

Since, the triangle is right-angled.

Statement III: III. The angles A, C and B of the triangle are in AP.

The angles are in Arithmetic Progression because:

This confirms that the angles are in AP.

Statement II is not correct because there is no mention of a side being 3 times the other.

∴ The correct answer is Option (I) and (III) are correct.

Hence, the correct answer is Option 3.

Top Properties of Triangles MCQ Objective Questions

If the three sides of Δ ABC are: a = 12 units, b = 14 units and c = 16 units. Find the value of cos B ? 

  1. 50/39
  2. 17/32
  3. 51/71
  4. None of these

Answer (Detailed Solution Below)

Option 2 : 17/32

Properties of Triangles Question 6 Detailed Solution

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Concept:

If three sides of Δ ABC are a, b and c units. Then

Calculation:

Here, the sides of the Δ ABC are a = 12 units, b = 14 units and c = 16 units.

We know that , by substituting the values of a, b and c in the formula, we get

⇒ 

Find the lengths of the sides of a triangle, if its angles are in the ratio 1: 2: 3 and the circum-radius is 10 cm.

  1. 5 cm, 6cm and 10 cm
  2. 10 cm, 10√3 cm and 20 cm
  3. 3 cm, 4 cm and 5 cm
  4. None of these

Answer (Detailed Solution Below)

Option 2 : 10 cm, 10√3 cm and 20 cm

Properties of Triangles Question 7 Detailed Solution

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Concept:

  • Sine Rule:

In a triangle Δ ABC, where a is the side opposite to A, b is the side opposite to B, c is the side opposite to C and where R is the circum-radius:

Calculation:

Given: Angles of a triangle are in the ratio 1: 2: 3 and the circum-radius is 10 cm.

Let A: B: C = 1: 2: 3 or A = k, B = 2k and C = 3k where k is any real number and R = 10 cm.

 As we know that, A + B + C = 180°

⇒ A + B + C = k + 2k + 3k = 180°

⇒ k = 30°

⇒ A = 30°, B = 60° and C = 90°

As we know that,

⇒ a = 20 × sin 30° = 10 cm, b = 20 × sin 60° = 10√3 cm and c = 20 × sin 90° = 20 cm.

If in a Δ ABC, the sides are a = 3 units, b = 5 units and c = 3 units. Find the value of cos A ?

  1. 5/6
  2. 1/2
  3. 4/5
  4. None of these

Answer (Detailed Solution Below)

Option 1 : 5/6

Properties of Triangles Question 8 Detailed Solution

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Concept:

For a Δ ABC, with sides a, b and c. 

 and 

Calculation:

Here, the sides of the Δ ABC are a = 3 units, b = 5 units and c = 3 units and we know that, .

⇒ 

⇒ cos A = 5/6

In equilateral ΔABC, D and E are points on the sides AB and AC, respectively, such that AD = CE. BE and CD intersect at F. The measure (in degrees) of ∠CFB is:

  1. 120° 
  2. 135° 
  3. 125° 
  4. 105° 

Answer (Detailed Solution Below)

Option 1 : 120° 

Properties of Triangles Question 9 Detailed Solution

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Given:

D and E are points on the sides AB and AC

AD = CE

BE and CD intersect at F

Concept used:

Concept of congruency of the triangle,

The exterior angle is always equal to the sum of the interior opposite angle. 

Calculation:

ΔCBE ≅ ΔACD [SAS congruency]

So, the three angles of these two triangles are the same,

Let ∠EBC be θ from this ∠ACD is also θ

Now,

∠BEC = 180° - (60° + θ)

⇒ 120° - θ

Now, In ΔECF

Exterior angle ∠CFB = (120° - θ) + θ

⇒ 120°

∴ ∠CFB is 120°

What is the measure of the two equal angles of a right isosceles triangle?

  1. 30°
  2. 45°
  3. 60°
  4. 90°

Answer (Detailed Solution Below)

Option 2 : 45°

Properties of Triangles Question 10 Detailed Solution

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Concept:

Angle sum property: The sum of angles of a triangle is 180°

Calculation:

Here, we have to find the measure of the two equal angles of a right isosceles triangle.

Let Δ ABC be a right isosceles triangle with ∠ B = 90° and AB = BC.

As we know, the angles opposite to equal sides are also equal.

⇒ ∠ACB = ∠BAC = x

Now by angle sum property, we have

⇒ x + x + 90° = 180°

⇒ x = 45°

Hence, the measure of the two equal angles of a right isosceles triangle is 45°.

Key Points

In a right isosceles triangle, one angle will be 90° and the other two sides will be equal.

The angles opposite to equal sides will also be equal.

If sin (C + D) = √3/2 and sec (C - D) = 2/3 then what is the value of C and D?

  1. 45 degree and 15 degree
  2. 30 degree and 30 degree
  3. 15 degree and 30 degree
  4. 60 degree and 30 degree

Answer (Detailed Solution Below)

Option 1 : 45 degree and 15 degree

Properties of Triangles Question 11 Detailed Solution

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Given:

 sin (C + D) = √3/2

sec (C - D) = 2/√3

Calculations:

If  sin (C + D) = √3/2 and sec (C - D) = 2/√3

Then,

⇒ C + D = 60°.............(1)

C - D = 30°..............(2)

Solving 1 & 2 .

C = 45°

D = 15°

∴ Option 1 is the correct answer.

The sides of a triangle are m, n and . What is the sum of the acute angles of the triangle?

  1. 45°
  2. 60°
  3. 75°
  4. 90°

Answer (Detailed Solution Below)

Option 2 : 60°

Properties of Triangles Question 12 Detailed Solution

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Concept:

Cosine Rule of Triangle:

The square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

Consider, a, b, and c are lengths of the side of a triangle ABC as shown, then;

Calculation:

Let m = n = 1 unit

Then,

  

Using cosine rule;

⇒ cos θ = -1/2

∴ θ = 120° 

Now, the sum of the acute angles of the triangle = 180° - 120° = 60° 

In a ΔABC, if a = 13, b = 14 and c = 15 then find the value of tan (C/2) ?

  1. 1/3
  2. 2/3
  3. 4/3
  4. None of these

Answer (Detailed Solution Below)

Option 2 : 2/3

Properties of Triangles Question 13 Detailed Solution

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Concept:

If a, b and c are the sides of the Δ ABC such that, a + b + c = 2S then

  •  

 

Calculation:

Given: For ΔABC we have a = 13, b = 14 and c = 15

Here, we have to find the value of tan (C/2)

As we know that, if a, b and c are the sides of the Δ ABC then 2S = a + b + c

⇒ 2S = 13 + 14 + 15 = 42

⇒ S = 21

As we know that, 

⇒ 

Hence, option 2 is the correct answer.

In a ΔABC, if a = 18, b = 24 and c = 30 then find the value of sin (A/2) ?

  1. None of these

Answer (Detailed Solution Below)

Option 1 :

Properties of Triangles Question 14 Detailed Solution

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CONCEPT:

If a, b and c are the sides of the Δ ABC such that, a + b + c = 2S then 

CALCULATION:

Given: For ΔABC we have a = 18, b = 24 and c = 30

Here, we have to find the value of  sin (A/2)

As we know that, if a, b and c are the sides of the Δ ABC then 2S = a + b + c

⇒ 2S = 18 + 24 + 30 = 72

⇒ S = 36

As we know that, 

Hence, option A is the correct answer.

In a triangle ABC, sec A (sin B cos C + cos B sin C) equals:

  1. c/a
  2. 1
  3. tan A
  4. cot A

Answer (Detailed Solution Below)

Option 3 : tan A

Properties of Triangles Question 15 Detailed Solution

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Concept:

  • sin (A + B) = sin A cos B + cos A sin B  ___(1)
  • The sum of the three angles of a triangle is 180° 
  • sin(180 - θ) = sin θ  ___(2)

Calculation:

In triangle ABC, sum of angles = A + B + C = 180° 

⇒ B + C = 180 - A   ___(3)

Given, sec A (sin B cos C + cos B sin C)

⇒ sec A sin (B + C)     ∵ {Using (1)}

⇒ sec A. sin (180 - A)   ∵ {Using (2)}

⇒ sec A. sin A   ∵ {Using (3)}

⇒  

⇒ tan A

∴ The correct answer is option (3).

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