Question

List the angles of the triangle in order from smallest to largest. In triangle Upper A Upper C Upper B, line segment Upper A Upper C has length 4.3, line segment Upper B Upper C has length 5.9, and line segment Upper B Upper A has length 3.5. A C B 5.9 4.3 3.5 Choose the correct order of the angles from smallest to largest.

Answer 1

`C = 36.0`

`B = 46.2`

`A = 97.8`

Explanation:

Given

`\triangle ABC`

`AC = 4.3`

`BC = 5.9`

`BA = 3.5`

Required

List the angles from smallest to largest

The given parameters is illustrated with the attached image.

`AC = 4.3` -- b

`BC = 5.9` --- a

`BA = 3.5` --- c

This question will be solved using cosine rule

To calculate A, we have:

`a^2 = b^2 + c^2 -2bc\ cos(A)`

So, we have:

`5.9^2 = 4.3^2 + 3.5^2 - 2 * 4.3 * 3.5 * \cos(A)`

`34.81 = 18.49+ 12.25 - 30.10* \cos(A)`

Collect like terms

`34.81 - 18.49- 12.25 = - 30.10* \cos(A)`

`4.07 = - 30.10* \cos(A)`

Make cos(A) the subject

`\cos(A) = -(4.07)/(30.10)`

`\cos(A) = -0.1352`

Take arccos of both sides

`A = cos^(-1)(-0.1352)`

`A = 97.8`

Solving for B, we have:

`b^2 = a^2 + c^2 -2ac\ cos(B)`

This gives:

`4.3^2 = 5.9^2 + 3.5^2 -2*5.9*3.5\ cos(B)`

`18.49 = 34.81+ 12.25 -41.30 *\cos(B)`

Collect like terms

`18.49 - 34.81 - 12.25 = -41.30 *\cos(B)`

`-28.57 = -41.30 *\cos(B)`

Solve for cos(B)

`\cos(B) = (-28.57)/(-41.30)`

`\cos(B) = 0.6918`

Take arccos of both sides

`B = cos^(-1)(0.6918)`

`B = 46.2`

To solve for C, we make use of:

`A + B + C = 180` --- angles in a triangle

`97.8 + 46.2 + C = 180`

Collect like terms

`C = - 97.8 - 46.2 + 180`

`C = 36.0`

So, we have:

`C = 36.0`

`B = 46.2`

`A = 97.8`