Determine whether these three side lengths will form a triangle: 6 cm, 8 cm, and 10 cm.(1 point) Responses They will form a triangle because the sum of any two of the side lengths is less than the length of the third side. They will form a triangle because the sum of any two of the side lengths is less than the length of the third side. They cannot form a triangle because the sum of any two of the side lengths is greater than the length of the third side. They cannot form a triangle because the sum of any two of the side lengths is greater than the length of the third side. They cannot form a triangle because the sum of any two of the side lengths is less than the length of the third side. They cannot form a triangle because the sum of any two of the side lengths is less than the length of the third side. They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side.

Answer 1

To determine whether the side lengths of 6 cm, 8 cm, and 10 cm can form a triangle, we can use the triangle inequality theorem. This theorem states that for any three side lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be met:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Now, let's apply these conditions to the given side lengths:

• Let \(a = 6\), \(b = 8\), and \(c = 10\).

1. \(6 + 8 > 10\) --> \(14 > 10\) (true)
2. \(6 + 10 > 8\) --> \(16 > 8\) (true)
3. \(8 + 10 > 6\) --> \(18 > 6\) (true)

All three conditions are satisfied. Therefore, these side lengths will indeed form a triangle.

The correct response would be: "They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side."

Answer 2

Based on the Triangle Inequality Theorem, which of the following types of triangles is possible?(1 point) Responses right isosceles right isosceles obtuse equilateral obtuse equilateral right equilateral right equilateral equilateral scalene

Answer 3

To determine which types of triangles are possible based on the Triangle Inequality Theorem, we need to understand the characteristics of the different types of triangles:

1. Right Triangle: A triangle with one angle measuring 90 degrees. It is possible as long as the sides satisfy the Pythagorean theorem (one side squared plus the other side squared equals the hypotenuse squared).

2. Isosceles Triangle: A triangle with at least two sides of equal length. It is possible as long as the lengths satisfy the triangle inequality.

3. Obtuse Triangle: A triangle with one angle greater than 90 degrees. It is allowed as long as the sides still satisfy the triangle inequality.

4. Equilateral Triangle: A triangle where all three sides are equal. It automatically satisfies the triangle inequality since all sides are equal.

5. Scalene Triangle: A triangle with all sides of different lengths. It can exist as long as the triangle inequality is fulfilled.

Based on the Triangle Inequality Theorem, all types of triangles listed (right, isosceles, obtuse, equilateral, and scalene) are indeed possible, provided the side lengths chosen satisfy the triangle inequality.

Given your responses, none of them specifically asserts that all types are possible, but you can consider that:

• Right triangles and isosceles triangles are specifically mentioned as possible without restrictions.
• Similarly, obtuse, equilateral, and scalene triangles are also part of the permissible types.

To summarize: The correct conclusion is that all types of triangles mentioned are possible under the Triangle Inequality Theorem, provided that their specific side lengths satisfy the triangle inequality.

Answer 4

which is the answer

Answer 5

Based on the options you provided, the simplest way to answer would be to identify one or more options that state a type of triangle is possible according to the Triangle Inequality Theorem.

Assuming your responses included specific combinations and you want a single answer:

• Any type of triangle mentioned in your responses is indeed possible.

However, if you are looking for a triangle type that is synonymous with the conditions of the Triangle Inequality Theorem, you can choose isosceles or equilateral.

If you took multiple-choice answers, you might want to select:

• Equilateral (since it is always valid as all sides are equal)
• Or Right or Isosceles (as they can also meet the triangle's requirements based on lengths).

If further clarification or more specific options were inadvertently left out, please provide those for tailored assistance!

Answer 6

A triangle’s side lengths are whole numbers. If the measure of two of its sides are 1 cm and 13 cm, classify the triangle by its sides.(1 point) Responses scalene scalene acute acute isosceles isosceles equilateral

Answer 7

To classify a triangle based on its side lengths, we can use the triangle inequality theorem, which states that for any triangle with sides \( a \), \( b \), and \( c \), the following must hold:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Given two sides measuring 1 cm and 13 cm, we need to find the possible length of the third side to determine if a triangle can be formed and how to classify it.

Let’s call the third side \( x \). According to the triangle inequality conditions, we have:

1. \( 1 + 13 > x \)

   • \( 14 > x \) (or \( x < 14 \))

2. \( 1 + x > 13 \)

   • \( x > 12 \)

3. \( 13 + x > 1 \)

   • This condition is always true for positive \( x \) since \( 13 + x > 1 \).

From these conditions, we can combine the inequalities:

• \( 12 < x < 14 \)

Since \( x \) must be a whole number, the only possible value for \( x \) is 13. This gives us side lengths of 1 cm, 13 cm, and 13 cm.

Classification by sides:

• Isosceles Triangle: Since two sides (13 cm and 13 cm) are equal.

Final Answer:

The triangle can be classified as isosceles.