Question

Hello good day can you help me please:(​

Answer 1

Answer:

  39.  C. 81°

  40.  A. 5678 ft²

Step-by-step explanation:

For solving a triangle from side lengths, the Law of Cosines is most useful. When an angle is known the area formula using that angle is applicable. If only side lengths are known, then Heron's formula is useful for finding the area.

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39.

The largest angle is opposite the longest side. The law of cosines tells us ...

  c² = a² +b² -2ab·cos(C)

Solving for the angle, we find ...

  C = arccos((a² +b² -c²)/(2ab)) = arccos((100² +115² -140²)/(2·100·115))

  C = arccos(3625/23000) ≈ 80.93°

The largest angle is about 81°.

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40.

The area of the triangle can be found using the formula ...

  Area = 1/2ab·sin(C)

  Area = 1/2(100)(115)sin(80.93188°) ≈ 5678.13 . . . square feet

The area of the triangular playground is about 5678 square feet.

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Additional comment

The largest angle of a triangle cannot be less than 60°, so we only need to determine whether the triangle is obtuse. The positive sign of cos(C) tells us it is not. That only leaves one answer choice for the angle.

We know a right triangle would have sides of approximately 100, 100, 141, so the given side lengths suggest the triangle is acute, rather than obtuse.

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Numerous triangle solvers are available online and as apps. The attachment shows the output of one of them for this triangle.