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

Explanation:

Given,

⇒ 

⇒b²+ c ² – a ² = c ² + a ² – b ² = a ² + b ² – c ²

⇒ a ² = b ² = c ²  

⇒ a = b = c

Hence, ABC is an equilateral triangle

Side of the equilateral triangle, a = 6 cm

The area of the equilateral triangle is,

⇒

⇒

⇒

∴ The area of the triangle is 9 × √3 square cm.

Calculation

Let the sides be .

It is understood that d > 0\) and from the figure, is greatest and is smallest.

By given condition, .

And hence .

Hence by sine rule we have,

or .

and .

.

.

or .

sides are

or .

Hence the required ratio is .

Then the sum of ratio of its sides is  = 15

Concept

sin3x = 3sinx - 4sin³x

sin2x = 1 - cos2x

In a Δ ABC 

Calculation

2b2 = a2 + c2

⇒ 3 - 4sin² B

⇒ 3 - 4(1 - cos2 B)

⇒ 4cos2 B - 1

⇒ 

⇒ 

⇒ 

⇒ 

Hence option 4 is correct

Calculation

In

, and

Hence option 3 is correct

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

⇒ 

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.

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

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°

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.

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.

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° 

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.

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.

Concepts:

• Sine law ⇔

Where a, b and c are sides and A, B and C are angles.

Here; Side a faces angle A, side b faces angle B and side c faces angle C

Calculation:

Given: a = 2, b = 3 and sin A = 2/3

Apply sine rule,

∴ B = π/2