Question

By first calculating the angle of LMN, calculate the area of triangle MNL. You must show all your working.

Answer 1

The Area of ∆MNL is 16.66cm².

∆LMN where mN = 38°

Side length NL = 7.2cm

ML = 4.8cm side length

Required:

∆ MNL Region

Solution:

Step 1: Using the sine rule, find Angle LMN sin(A)/a = sin(B)/b.

In which case sin(A) = sin(M) =?

a = NL = 7.2cm

38° sin(B) = sin(N)

b = ML = 4.8cm

Thus,

Sin(M)/7.2 equals sin(38)/4.8.

Multiply by 2

4.8 * sin(M) = 7.2 * sin(38)

4.8*sin(M) = 7.2*0.6157

4.8*sin(M) = 4.43304

Multiply both sides by 4.8.

sin(M) = 4.43304/4.8

sin(M) = 0.92355

M = sin-¹(0.92355) ≈ 67.45°

Step 2: Find m<L

angle M + angle N + angle L = 180 (sum of triangle angles)

180 = 67.45 + 38 + angle L

180 = 105.45 + angle L

Take 105.45 off both sides.

L = 180 minus 105.45

L angle = 74.55°

Step 3: Using the formula 12*a*b*sin(C), calculate the area of MNL.

Where,

a = NL = 7.2 cm

b = ML = 4.8 cm

sin(C) = sin(L) + sin(74.55) = sin(74.55)

Thus,

MNL area = 12*7.2*4.8*0.9639

= ½*33.31

= 16.655

Area of ∆MNL ≈ 16.66cm²

Answer 2

16.66cm²

Explanation:

Given:

∆LMN with m<N = 38°

Length of side NL = 7.2cm

Length of side ML = 4.8cm

Required:

Area of ∆MNL

Solution:

Step 1: Find Angle LMN using the sine rule sin(A)/a = sin(B)/b

Where sin(A) = Sin(M) = ?

a = NL = 7.2cm

sin(B) = sin(N) = 38°

b = ML = 4.8cm

Thus,

Sin(M)/7.2 = sin(38)/4.8

Cross multiply

4.8*sin(M) = 7.2*sin(38)

4.8*sin(M) = 7.2*0.6157

4.8*sin(M) = 4.43304

Divide both sides by 4.8

sin(M) = 4.43304/4.8

sin(M) = 0.92355

M = sin-¹(0.92355) ≈ 67.45°

Step 2: Find m<L

angle M + angle N + angle L = 180 (sum of angles in a triangle)

67.45 + 38 + angle L = 180

105.45 + angle L = 180

Subtract 105.45 from both sides

Angle L = 180 - 105.45

Angle L = 74.55°

Step 3: Find the area of ∆MNL using the formula ½*a*b*sin(C)

Where,

a = NL = 7.2 cm

b = ML = 4.8 cm

sin(C) = sin(L) = sin(74.55)

Thus,

Area of ∆MNL = ½*7.2*4.8*0.9639

= ½*33.31

= 16.655

Area of ∆MNL ≈ 16.66cm²