Here are the answers to your questions:
Question 1: To determine whether the three side lengths (6 cm, 8 cm, and 10 cm) can form a triangle, we apply the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
- 6 + 8 = 14 > 10 (True)
- 6 + 10 = 16 > 8 (True)
- 8 + 10 = 18 > 6 (True)
Since all three conditions are met, the correct response is:
They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side.
Question 2: Using the Triangle Inequality Theorem again, we have two pieces measuring 12 inches and 16 inches. Let's denote the length of the third piece as x. The following inequalities must hold:
- 12 + 16 > x → 28 > x (or x < 28)
- 12 + x > 16 → x > 4
- 16 + x > 12 → x > -4 (this is always true since x is positive)
Combining the inequalities:
4 < x < 28
Thus, the valid options for the length of the third piece are anything greater than 4 inches and less than 28 inches. Among the provided options, the only valid length is:
4 inches.
(This piece of wood can't be exactly 4 inches without violating the triangle formation requirement since it must be greater than 4). Thus, the previous value provided is misleading in format but must be accurately stated as a range: 4 < x < 28.
Question 3: For a triangle to exist, it must satisfy the Triangle Inequality Theorem. All of the options imply valid combinations of triangle sides.
- Obtuse equilateral triangles exist as they adhere to the triangle inequality properties.
- Equilateral scalene is incorrect because, by definition, an equilateral triangle has all sides equal.
- Right isosceles triangles are valid.
- Right equilateral is not a valid classification as equilateral implies all sides are equal.
Thus, the correct answer is:
obtuse equilateral.
Question 4: For two sides measuring 1 cm and 13 cm, you can check if a triangle can be formed:
The lengths must satisfy the triangle inequality:
- 1 + 13 > x (for the third side)
- 1 + x > 13 (x must be greater than 12)
- 13 + x > 1 (this is always true since x is positive).
Since x must satisfy x > 12, there's only one possible whole number for the third side, which excludes making it equilateral, isosceles, or acute.
Thus, it can only be classified as:
scalene.
Question 5: To find the range for the third side (let’s call it x) with the other two sides being 42.7 mm and 38.03 mm, we use the triangle inequality:
- 42.7 + 38.03 > x → 80.73 > x (or x < 80.73)
- 42.7 + x > 38.03 → x > -4.67 (this is always true since x is positive)
- 38.03 + x > 42.7 → x > 4.67
Thus, the combined conditions yield:
4.67 < x < 80.73.
So the final responses are:
- They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side.
- 4 inches.
- obtuse equilateral.
- scalene.
- 4.67 < x < 80.73.