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Sam is drawing triangles on banners using a ruler and protractor.

Inline Dropdown Question
On the second banner Sam wants to create a variety of different sizes of triangles.

For each triangle condition listed, determine whether the condition will produce one and only one triangle, more than one triangle, or no triangle.

Use the drop-down menu to select the appropriate answer.

A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches
Query 1
A triangle with side lengths of 5 inches, 7 inches, and 10 inches
Query 2
A triangle with each angle measuring 60 degrees
Query 3
A triangle with angle measures of 50, 50, and 60
Query 4

1 answer

To determine the number of triangles that can be formed based on the conditions provided, we will apply the triangle inequality theorem and the properties of triangles:

  1. A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches

    • The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side.
      • 4 + 5 = 9 (not greater than the third side)
    • This means that no triangle can be formed.
    • Answer: No triangle

  2. A triangle with side lengths of 5 inches, 7 inches, and 10 inches

    • Checking the triangle inequality:
      • 5 + 7 = 12 (greater than 10)
      • 5 + 10 = 15 (greater than 7)
      • 7 + 10 = 17 (greater than 5)
    • All conditions satisfied, so one triangle can be formed.
    • Answer: One and only one triangle

  3. A triangle with each angle measuring 60 degrees

    • A triangle with all angles equal (60 degrees) is an equilateral triangle, which is unique in size and shape if only the angles are given.
    • Answer: One and only one triangle

  4. A triangle with angle measures of 50, 50, and 60

    • The sum of the angles is 180 degrees, and having two angles equal (50 and 50) indicates it's an isosceles triangle, which has a unique shape and size for these angle measures.
    • Answer: One and only one triangle

In summary:

  • Query 1: No triangle
  • Query 2: One and only one triangle
  • Query 3: One and only one triangle
  • Query 4: One and only one triangle