Question
A) Determine whether these three side lengths will form a triangle: 6 cm, 8 cm, and 10 cm.
(1 point)
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 will 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.
(1 point)
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 will 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.
Answers
Answered by
GPT-4o mini
To determine whether the three 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 triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's check the pairs:
1. **6 cm + 8 cm > 10 cm**
\(14 > 10\) (True)
2. **6 cm + 10 cm > 8 cm**
\(16 > 8\) (True)
3. **8 cm + 10 cm > 6 cm**
\(18 > 6\) (True)
Since all three conditions are satisfied, the side lengths **will form a triangle**.
Therefore, the correct answer 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.**
Let's check the pairs:
1. **6 cm + 8 cm > 10 cm**
\(14 > 10\) (True)
2. **6 cm + 10 cm > 8 cm**
\(16 > 8\) (True)
3. **8 cm + 10 cm > 6 cm**
\(18 > 6\) (True)
Since all three conditions are satisfied, the side lengths **will form a triangle**.
Therefore, the correct answer 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.**
Answer
A) Alisha wants to build a miniature bridge that will include a triangle made of wooden pieces.
She has two wooden pieces, one of which measures 12 inches and the other of which
measures 16 inches. Which could be the length of the third piece of wood in order for her to
form a triangle? (1 point)
27 inches
28 inches
4 inches
3 inches
She has two wooden pieces, one of which measures 12 inches and the other of which
measures 16 inches. Which could be the length of the third piece of wood in order for her to
form a triangle? (1 point)
27 inches
28 inches
4 inches
3 inches
Answered by
GPT-4o mini
To determine the possible length of the third piece of wood that will allow Alisha to form a triangle with the two given pieces (12 inches and 16 inches), we can use the triangle inequality theorem. According to this theorem, the sum of any two sides of a triangle must be greater than the length of the third side.
Let's denote the lengths of the three sides as follows:
- \( a = 12 \) inches
- \( b = 16 \) inches
- \( c = \) length of the third piece of wood
We need to apply the triangle inequalities:
1. \( a + b > c \)
\( 12 + 16 > c \)
\( 28 > c \)
Therefore, \( c < 28 \)
2. \( a + c > b \)
\( 12 + c > 16 \)
\( c > 16 - 12 \)
\( c > 4 \)
3. \( b + c > a \)
\( 16 + c > 12 \)
\( c > 12 - 16 \)
Since this is always true for positive values of \( c \), we don't get any additional constraints here.
Now, combining the two useful conditions we have:
- \( c < 28 \)
- \( c > 4 \)
Thus, \( c \) must be greater than 4 inches and less than 28 inches.
Now, let's evaluate the given options:
- **27 inches**: Valid (because \( 4 < 27 < 28 \))
- **28 inches**: Not valid (because \( c \) must be less than 28)
- **4 inches**: Not valid (because \( c \) must be greater than 4)
- **3 inches**: Not valid (because \( c \) must be greater than 4)
Therefore, the only feasible length for the third piece of wood is **27 inches**.
Let's denote the lengths of the three sides as follows:
- \( a = 12 \) inches
- \( b = 16 \) inches
- \( c = \) length of the third piece of wood
We need to apply the triangle inequalities:
1. \( a + b > c \)
\( 12 + 16 > c \)
\( 28 > c \)
Therefore, \( c < 28 \)
2. \( a + c > b \)
\( 12 + c > 16 \)
\( c > 16 - 12 \)
\( c > 4 \)
3. \( b + c > a \)
\( 16 + c > 12 \)
\( c > 12 - 16 \)
Since this is always true for positive values of \( c \), we don't get any additional constraints here.
Now, combining the two useful conditions we have:
- \( c < 28 \)
- \( c > 4 \)
Thus, \( c \) must be greater than 4 inches and less than 28 inches.
Now, let's evaluate the given options:
- **27 inches**: Valid (because \( 4 < 27 < 28 \))
- **28 inches**: Not valid (because \( c \) must be less than 28)
- **4 inches**: Not valid (because \( c \) must be greater than 4)
- **3 inches**: Not valid (because \( c \) must be greater than 4)
Therefore, the only feasible length for the third piece of wood is **27 inches**.
Answer
A) Based on the Triangle Inequality Theorem, which of the following types of triangles is
possible? (1 point)
obtuse equilateral
right isosceles
equilateral scalene
right equilateral
possible? (1 point)
obtuse equilateral
right isosceles
equilateral scalene
right equilateral
Answered by
GPT-4o mini
To determine which types of triangles are possible based on the Triangle Inequality Theorem, we can analyze the characteristics of the triangles mentioned:
1. **Obtuse Equilateral Triangle**
- An equilateral triangle has all three sides equal and all angles equal to 60 degrees. Therefore, it's not possible for an equilateral triangle to be obtuse because it can't have any angle greater than 90 degrees.
- **Not possible.**
2. **Right Isosceles Triangle**
- An isosceles triangle has at least two sides equal. In a right isosceles triangle, the two equal sides form the right angle, which makes it possible as it satisfies the triangle inequality.
- **Possible.**
3. **Equilateral Scalene Triangle**
- An equilateral triangle cannot be scalene because scalene triangles have all sides of different lengths.
- **Not possible.**
4. **Right Equilateral Triangle**
- Similar to the obtuse equilateral triangle case, an equilateral triangle cannot have a right angle since all angles are 60 degrees.
- **Not possible.**
Based on this analysis, the only possible type of triangle from the given options is:
**Right Isosceles.**
1. **Obtuse Equilateral Triangle**
- An equilateral triangle has all three sides equal and all angles equal to 60 degrees. Therefore, it's not possible for an equilateral triangle to be obtuse because it can't have any angle greater than 90 degrees.
- **Not possible.**
2. **Right Isosceles Triangle**
- An isosceles triangle has at least two sides equal. In a right isosceles triangle, the two equal sides form the right angle, which makes it possible as it satisfies the triangle inequality.
- **Possible.**
3. **Equilateral Scalene Triangle**
- An equilateral triangle cannot be scalene because scalene triangles have all sides of different lengths.
- **Not possible.**
4. **Right Equilateral Triangle**
- Similar to the obtuse equilateral triangle case, an equilateral triangle cannot have a right angle since all angles are 60 degrees.
- **Not possible.**
Based on this analysis, the only possible type of triangle from the given options is:
**Right Isosceles.**
Answer
A) 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)
isosceles
scalene
acute
equilateral
13 cm, classify the triangle by its sides. (1 point)
isosceles
scalene
acute
equilateral
Answered by
GPT-4o mini
To classify the triangle based on the lengths of its sides (1 cm and 13 cm) and determine whether a third side can exist that satisfies the triangle inequality theorem, we need to check the possible range for the third side.
According to the triangle inequality theorem, for any triangle with side lengths \( a \), \( b \), and \( c \):
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Given:
- Two sides are \( a = 1 \) cm and \( b = 13 \) cm.
We can denote the third side as \( c \). Now, we will use the triangle inequalities:
1. **From \( a + b > c \)**:
\[
1 + 13 > c \quad \Rightarrow \quad 14 > c \quad \Rightarrow \quad c < 14
\]
2. **From \( a + c > b \)**:
\[
1 + c > 13 \quad \Rightarrow \quad c > 12
\]
3. **From \( b + c > a \)**:
This condition will always be satisfied because \( 13 + c > 1 \) is true as long as \( c \) is positive.
Combining the constraints from the first two inequalities, we find that:
\[
12 < c < 14
\]
Thus, the only whole number that satisfies this inequality is \( c = 13 \).
Now, if we consider the sides of the triangle after determining \( c \):
- The sides are 1 cm, 13 cm, and 13 cm.
Since one side (13 cm) is repeated, the triangle is classified as **isosceles**.
Therefore, the answer is:
**Isosceles.**
According to the triangle inequality theorem, for any triangle with side lengths \( a \), \( b \), and \( c \):
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Given:
- Two sides are \( a = 1 \) cm and \( b = 13 \) cm.
We can denote the third side as \( c \). Now, we will use the triangle inequalities:
1. **From \( a + b > c \)**:
\[
1 + 13 > c \quad \Rightarrow \quad 14 > c \quad \Rightarrow \quad c < 14
\]
2. **From \( a + c > b \)**:
\[
1 + c > 13 \quad \Rightarrow \quad c > 12
\]
3. **From \( b + c > a \)**:
This condition will always be satisfied because \( 13 + c > 1 \) is true as long as \( c \) is positive.
Combining the constraints from the first two inequalities, we find that:
\[
12 < c < 14
\]
Thus, the only whole number that satisfies this inequality is \( c = 13 \).
Now, if we consider the sides of the triangle after determining \( c \):
- The sides are 1 cm, 13 cm, and 13 cm.
Since one side (13 cm) is repeated, the triangle is classified as **isosceles**.
Therefore, the answer is:
**Isosceles.**
Answer
A) Find the range of values for the third side of a triangle if two of its sides measure 42.7 mm
and 38.03 mm. (1 point)
4.67 < x < 42.3
4.67 < x < 80.73
4.4 < x < 80.73
4.04 < x < 80.1
and 38.03 mm. (1 point)
4.67 < x < 42.3
4.67 < x < 80.73
4.4 < x < 80.73
4.04 < x < 80.1
Answered by
GPT-4o mini
To find the range of values for the third side of a triangle given two sides measuring \( a = 42.7 \) mm and \( b = 38.03 \) mm, we will use the triangle inequality theorem.
According to the triangle inequality, for any triangle with side lengths \( a \), \( b \), and \( c \) (where \( c \) is the third side), the following must hold true:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
### Applying the inequalities:
1. **From \( a + b > c \)**:
\[
42.7 + 38.03 > c \\
80.73 > c \\
\text{or} \\
c < 80.73
\]
2. **From \( a + c > b \)**:
\[
42.7 + c > 38.03 \\
c > 38.03 - 42.7 \\
c > -4.67 \\
\text{(which is always true for positive values of \( c \))}
\]
3. **From \( b + c > a \)**:
\[
38.03 + c > 42.7 \\
c > 42.7 - 38.03 \\
c > 4.67
\]
### Combining the restrictions:
From the two relevant inequalities derived, we have:
- \( c > 4.67 \)
- \( c < 80.73 \)
Thus, the range of values for the third side \( c \) is:
\[
4.67 < c < 80.73
\]
### Conclusion:
The correct range of values for the third side of the triangle is:
**4.67 < x < 80.73.**
According to the triangle inequality, for any triangle with side lengths \( a \), \( b \), and \( c \) (where \( c \) is the third side), the following must hold true:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
### Applying the inequalities:
1. **From \( a + b > c \)**:
\[
42.7 + 38.03 > c \\
80.73 > c \\
\text{or} \\
c < 80.73
\]
2. **From \( a + c > b \)**:
\[
42.7 + c > 38.03 \\
c > 38.03 - 42.7 \\
c > -4.67 \\
\text{(which is always true for positive values of \( c \))}
\]
3. **From \( b + c > a \)**:
\[
38.03 + c > 42.7 \\
c > 42.7 - 38.03 \\
c > 4.67
\]
### Combining the restrictions:
From the two relevant inequalities derived, we have:
- \( c > 4.67 \)
- \( c < 80.73 \)
Thus, the range of values for the third side \( c \) is:
\[
4.67 < c < 80.73
\]
### Conclusion:
The correct range of values for the third side of the triangle is:
**4.67 < x < 80.73.**