Question

The angles of a triangle are in the ratio 11 : 5 : 2. The side opposite the largest angle is 5.5 centimeters. The length of the side opposite the smallest angle of the triangle is blank centimeters.

Answer 1

Step-by-step explanation:

a:b:c=11:5:2

= 11x:5x:2x

.

.

a+ b+c = 180° ( angle sum property )

11x + 5x + 2x = 180°

18x = 180°

x = 180/18

therefore, x= 10

.

.

a = 11x = 11 × 10 = 110°

b = 5x = 5 × 10 = 50°

c = 2x = 2 × 10 = 20°

.

.

Refer to the attachment

Answer 2

The length of the side opposite to the smallest angle of the triangle is 1.99 cm or ≈2 cm.

Given:

• The angles of a triangle are in the ratio 11 : 5 : 2.
• The side opposite the largest angle is 5.5 centimeters.

To find:

• Find the length of the side opposite the smallest angle of the triangle in cm

Solution:

Formula/concept to be used:

• Angle sum property of the triangle.
• Apply sine rule.

Step 1:

Calculate the angles of the triangle.

Let the common factor of the ratio be x.

So,

$11x + 5x + 2x = 180^{\circ}$

$18x = 180^{\circ}$

$\bf x =10 ^{\circ}$

So,

The angles of the triangle are 110°,50°, and 20°.

Step 2:

Find the length of the smallest side.

We know that, Side opposite to large angle is longest and the side opposite to small angle is smallest.

So,

Side opposite to 20° is the smallest side.

*See the attachment.

Apply sine rule in ∆ABC.

$\bf \frac{sin \: 110^{\circ}}{5.5} = \frac{sin \: 20^{\circ}}{CB} ...eq1$

Find the value of sin 110° and sin 20° form the table.

sin 110°= 0.94

sin 20°= 0.34

Put the values in the eq1.

$\frac{0.94}{5.5} = \frac{0.34}{CB}$

$CB = \frac{5.5 \times 0.34}{0.94}$

$\bf CB = 1.99 \: cm$

Thus,

The length of the smallest side of the triangle is 1.99 cm or ≈2 cm.

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