Question

If triangle abc has the following measurements, find the measure of side c. a = 10 b = 3 c = 15° 6.7 6.5 7.1 7.4

A) 6.7
B) 6.5
C) 7.1
D) 7.4

Answer 1

Using the law of cosines and the given measurements of a triangle, the measure of side c is calculated and approximated to be 7.1. Thus (option C) is right answer.

Step-by-step explanation:

To find the measure of side c in triangle ABC with measurements a = 10, b = 3, and C = 15°, we will use the law of cosines. This formula is c = √(a² + b² - 2ab·cos(C)). Plugging the values into the equation:

c = √(10² + 3² - 2·10·3·cos(15°))
= √(100 + 9 - 60·cos(15°))
= √(109 - 60·cos(15°))

Without a calculator, exact calculation can be tricky; however, by approximation and knowing that cos(15°) is roughly 0.96,

c ≈ √(109 - 60·0.96)
= √(109 - 57.6)
= √(51.4)
≈ 7.17

Looking at the choices provided, the closest to 7.17 is 7.1. Thus (option C) is right answer.

Answer 2

The measure of side c is approximately 7.2.

Apply the Law of Cosines

The Law of Cosines states that for any triangle ABC:

`\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]`

where c is the side opposite to angle C, and a and b are the other two sides.

In this case,
`\(a = 10\), \(b = 3\), and \(C = 15^\circ\),` so the equation becomes:

`\[ c^2 = 10^2 + 3^2 - 2 \cdot 10 \cdot 3 \cdot \cos(15^\circ) \]`

Evaluate the expression using a calculator:

`\[ c^2 = 100 + 9 - 60 \cdot \cos(15^\circ) \]`

`\[ c^2 = 109 - 60 \cdot 0.9659 \]`

`\[ c^2 = 109 - 57.95 \]`

`\[ c^2 = 51.95 \]`

`\[ c =√(51.95) \]`

c ≈ 7.217

So, the measure of side (c is approximately 7.2.