Calculate the length of ac

  12.1 cm

Step-by-step explanation:

Using the law of sines, we can find angle C. Then from the sum of angles, we can find angle B. The law of sines again will tell us the length AC.

  sin(C)/c = sin(A)/a

  C = arcsin((c/a)sin(A)) = arcsin(8.2/13.5·sin(81°)) ≈ 36.86°

Then angle B is ...

  B = 180° -A -C = 180° -81° -36.86° = 62.14°

and side b is ...

  b/sin(B) = a/sin(A)

  b = a·sin(B)/sin(A) = 13.5·sin(62.14°)/sin(81°) ≈ 12.0835

The length of AC is about 12.1 cm.

Comment on the solution

The problem can also be solved using the law of cosines. The equation is ...

  13.5² = 8.2² +b² -2·8.2·b·cos(81°)

This is a quadratic in b. Its solution can be found using the quadratic formula or by completing the square.

  b = 8.2·cos(81°) +√(13.5² -8.2² +(8.2·cos(81°))²)

  b = 8.2·cos(81°) +√(13.5² -(8.2·sin(81°))²) . . . . . simplified a bit

You can use the law of cosines. Let be the length of AC. Then

and you can use the quadratic formula to find . The solutions are approximately -9.518 and 12.084. Take the positive solution, since we're talking about length.

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