Question

A man claims that his lot is triangular, with one side 450 m long and the adjacent side 200 m long. The

angle opposite one side is 28º. Determine the other side length of this lot to the nearest metre.

Answer 1

289 m or 617 m

Explanation:

You want the third side length of a triangle with side lengths 450 m and 200 m, with an angle of 28°.

Solution 1

The man's claim does not say which side the given angle is opposite. There are two possibilities. (1) It is opposite the unknown side; (2) it is opposite the side of length 450 m. (No triangle is possible having an angle of 28° opposite the shorter given side.)

If the angle is opposite the unknown side, the law of cosines can be used to find the third side length:

c² = a² + b² - 2ab·cos(C)

c² = 450² +200² -2·450·200·cos(28°) ≈ 83569.43

c ≈ √83569.43 ≈ 289 . . . . meters

The other side length could be 289 meters.

Solution 2

The third side could also be figured using the law of sines.

a/sin(A) = b/sin(B) = c/sin(C)

450/sin(28°) = 200/sin(B)

B = arcsin(200/450·sin(28°)) ≈ 12.043°

Then angle C is ...

C = 180° -28° -12.043° = 139.957°

and side 'c' is ...

c = 450·sin(139.957°)/sin(28°) ≈ 617 . . . . meters

The other side length could be 617 meters.

__

Additional comment

The problem tells us "one side" is 450 m, and it tells us the angle opposite "one side" is 28°. If both of the descriptors "one side" are referring to the same side, then Solution 2 is the intended one.

The description can be written in a less ambiguous way. As is, we are not sure that the second use of "one side" is referring to any side in particular. Hence the two possibilities.

<95141404393>

Answer 2

• Accurate answer: 617 m

• Answer out of available options: C. 616 m

Explanation:

Given information:

• Side a = 450 m (opposite angle A)

• Side b = 200 m (opposite angle B)

• Angle A = 28°

We can use the Law of Sines to find angle B:

• 
  `$(a)/(sin(A)) = (b)/(sin(B))`

Substitute the given values:

• 
  `$(450)/(sin(28^\circ)) = (200)/(sin(B))`

Now, solve for angle B:

• 
  `$sin(B) = (200 * sin(28^\circ))/(450)`

• 
  `$sin(B) \approx 0.208654`

• 
  `$B \approx \arcsin{0.208654} \approx 12.0435^\circ`

Now that we have angle B, we can find angle C using the fact that the sum of the interior angles in a triangle is always 180°:

• Angle C = 180° - 28° - 12.0435°

• Angle C = 139.9565°

Now, we can use the Law of Sines again to find the length of the other side (side c) opposite angle C:

• 
  `$(a)/(sin(A)) = (c)/(sin(C))`

Substitute the given values:

• 
  `$(450)/(sin(28^\circ)) = (c)/(sin(139.9565^\circ))`

Now, solve for side c:

• 
  `$c = (450 * sin(139.9565^\circ))/(sin(28^\circ))`

• 
  `$c \approx 616.685`

To the nearest meter, the other side length of the triangular lot is approximately 617 m. But, since there is not an option for the answer, the closest option is C. 616 m.

________________________________________________________

Full Question