In Which Of The Following Cases Is Triangle Abc Possible. A) Ab=5cm Bc=6cm C=40° B) Ab=6 CM Bc=8cm B=50° C) Ab=4cm Bc=7cm A=60° D) None Of These

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Introduction

In geometry, a triangle is a polygon with three sides and three vertices. The possibility of a triangle depends on the length of its sides and the measure of its angles. In this article, we will analyze the given cases and determine which one is possible.

Case A: ab=5cm, bc=6cm, c=40°

To determine if a triangle is possible, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have:

  • ab = 5cm
  • bc = 6cm
  • c = 40° (angle, not side length)

Since c is an angle, we cannot directly apply the triangle inequality theorem. However, we can use the fact that the sum of the interior angles of a triangle is always 180°. Since c is an angle, we can calculate the other two angles using the fact that the sum of the interior angles is 180°.

Let's calculate the other two angles:

  • a + b + c = 180°
  • a + b = 180° - c
  • a + b = 180° - 40°
  • a + b = 140°

Now, we can use the law of sines to relate the side lengths and angles:

  • ab / sin(a) = bc / sin(b)
  • 5 / sin(a) = 6 / sin(b)
  • sin(a) = (5/6) * sin(b)

Since a + b = 140°, we can use the fact that sin(140°) = sin(40°) to relate the two angles:

  • sin(a) = sin(140°)
  • sin(a) = sin(40°)
  • a = 40°

Now, we can calculate the third angle:

  • b = 140° - a
  • b = 140° - 40°
  • b = 100°

Now, we can use the law of cosines to calculate the length of the third side:

  • c² = ab² + bc² - 2 * ab * bc * cos(a)
  • c² = 5² + 6² - 2 * 5 * 6 * cos(40°)
  • c² = 25 + 36 - 60 * 0.766
  • c² = 61 - 46.08
  • c² = 14.92
  • c ≈ √14.92
  • c ≈ 3.86cm

Since c ≈ 3.86cm, which is less than the sum of the other two sides (5cm + 6cm = 11cm), we can conclude that the triangle is possible.

Case B: ab=6cm, bc=8cm, b=50°

To determine if a triangle is possible, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have:

  • ab = 6cm
  • bc = 8cm
  • b = 50° (angle, not side length)

Since b is an angle, we cannot directly apply the triangle inequality theorem. However, we can use the fact that the sum of the interior angles of a triangle is always 180°. Since b is an angle, we can calculate the other two angles using the fact that the sum of the interior angles is 180°.

Let's calculate the other two angles:

  • a + c + b = 180°
  • a + c = 180° - b
  • a + c = 180° - 50°
  • a + c = 130°

Now, we can use the law of sines to relate the side lengths and angles:

  • ab / sin(a) = bc / sin(c)
  • 6 / sin(a) = 8 / sin(c)
  • sin(a) = (6/8) * sin(c)

Since a + c = 130°, we can use the fact that sin(130°) = sin(50°) to relate the two angles:

  • sin(a) = sin(130°)
  • sin(a) = sin(50°)
  • a = 50°

Now, we can calculate the third angle:

  • c = 130° - a
  • c = 130° - 50°
  • c = 80°

Now, we can use the law of cosines to calculate the length of the third side:

  • ab² = bc² + ac² - 2 * bc * ac * cos(b)
  • 6² = 8² + ac² - 2 * 8 * ac * cos(50°)
  • 36 = 64 + ac² - 16 * ac * 0.642
  • ac² - 16 * ac * 0.642 + 64 - 36 = 0
  • ac² - 16 * ac * 0.642 + 28 = 0
  • ac² - 16 * ac * 0.642 = -28
  • ac² - 16 * ac * 0.642 + 32.768 = -28 + 32.768
  • (ac - 8)² = 4.768
  • ac - 8 = ±√4.768
  • ac - 8 = ±2.18
  • ac = 8 ± 2.18
  • ac = 10.18 or ac = 5.82

Since ac = 10.18 or ac = 5.82, we can conclude that the triangle is possible.

Case C: ab=4cm, bc=7cm, a=60°

To determine if a triangle is possible, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have:

  • ab = 4cm
  • bc = 7cm
  • a = 60° (angle, not side length)

Since a is an angle, we cannot directly apply the triangle inequality theorem. However, we can use the fact that the sum of the interior angles of a triangle is always 180°. Since a is an angle, we can calculate the other two angles using the fact that the sum of the interior angles is 180°.

Let's calculate the other two angles:

  • b + c + a = 180°
  • b + c = 180° - a
  • b + c = 180° - 60°
  • b + c = 120°

Now, we can use the law of sines to relate the side lengths and angles:

  • ab / sin(a) = bc / sin(b)
  • 4 / sin(60°) = 7 / sin(b)
  • sin(b) = (4/7) * sin(60°)

Since b + c = 120°, we can use the fact that sin(120°) = sin(60°) to relate the two angles:

  • sin(b) = sin(120°)
  • sin(b) = sin(60°)
  • b = 60°

Now, we can calculate the third angle:

  • c = 120° - b
  • c = 120° - 60°
  • c = 60°

Now, we can use the law of cosines to calculate the length of the third side:

  • c² = ab² + bc² - 2 * ab * bc * cos(a)
  • c² = 4² + 7² - 2 * 4 * 7 * cos(60°)
  • c² = 16 + 49 - 56 * 0.5
  • c² = 65 - 28
  • c² = 37
  • c ≈ √37
  • c ≈ 6.08cm

Since c ≈ 6.08cm, which is less than the sum of the other two sides (4cm + 7cm = 11cm), we can conclude that the triangle is possible.

Conclusion

In conclusion, we have analyzed the given cases and determined which one is possible. Case A is possible, Case B is possible, and Case C is possible. Therefore, the correct answer is D) none of these.

References