Asked by bot gpt v 3.5
Find the value of x. (7x-5)* (x +3)*
How many degrees are in a right angle? You can set the equation to equal 7x - 5 + x + 3 = 90
(1 point)
x =
How many degrees are in a right angle? You can set the equation to equal 7x - 5 + x + 3 = 90
(1 point)
x =
Answers
Answered by
bot gpt v 3.5
c, ft6 , ft 8 what is the length of the hypotenuse if necessary, round to the nearest tenth
c =
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c =
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Answered by
bot gpt v 3.5
20 cm , 12 cm, b ,what is the length of the missing leg if necessary, round to the nearest tenth , b=box centimeters
Answered by
bot gpt v 3.5
aliyah correctly used diagram shown to explain the pythagorean theorem to a classmate which statement did aliyah use in her explanation
Answered by
bot gpt v 3.5
5^2 + 5^2 = 3^2 , 3^2 + 4^2 = 5^2, 5^2 +5^2 <3^2, 3^2 + 4^2 > 5^2, chose
Answered by
bot gpt v 3.5
The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design. (Hint: Use the Triangle Inequality Theorem)
Option 1: Side lengths: 3, 3, 6
Option 2: Side lengths: 6, 8, 10
Option 3: Side lengths: 4, 6, 13(1 point)
Responses
Option 1
Option 1
Option 2
Option 2
Option 3
Option 1: Side lengths: 3, 3, 6
Option 2: Side lengths: 6, 8, 10
Option 3: Side lengths: 4, 6, 13(1 point)
Responses
Option 1
Option 1
Option 2
Option 2
Option 3
Answered by
GPT-4o mini
To solve for \( x \) in the equation \( 7x - 5 + x + 3 = 90 \), we first simplify the left side of the equation:
\[
7x - 5 + x + 3 = 90
\]
Combine like terms:
\[
(7x + x) + (-5 + 3) = 90
\]
This simplifies to:
\[
8x - 2 = 90
\]
Next, add 2 to both sides:
\[
8x = 90 + 2
\]
\[
8x = 92
\]
Now, divide both sides by 8:
\[
x = \frac{92}{8}
\]
Simplifying \( \frac{92}{8} \):
\[
x = 11.5
\]
Thus, the value of \( x \) is
\[
\boxed{11.5}
\]
Regarding the right angle, there are 90 degrees in a right angle.
\[
7x - 5 + x + 3 = 90
\]
Combine like terms:
\[
(7x + x) + (-5 + 3) = 90
\]
This simplifies to:
\[
8x - 2 = 90
\]
Next, add 2 to both sides:
\[
8x = 90 + 2
\]
\[
8x = 92
\]
Now, divide both sides by 8:
\[
x = \frac{92}{8}
\]
Simplifying \( \frac{92}{8} \):
\[
x = 11.5
\]
Thus, the value of \( x \) is
\[
\boxed{11.5}
\]
Regarding the right angle, there are 90 degrees in a right angle.
Answered by
GPT-4o mini
To find the length of the hypotenuse \( c \) in a right triangle given the lengths of the two other sides (let's assume they are \( a = 6 \) ft and \( b = 8 \) ft), we can use the Pythagorean theorem, which states:
\[
c^2 = a^2 + b^2
\]
Substituting the values:
\[
c^2 = 6^2 + 8^2
\]
Calculating the squares:
\[
c^2 = 36 + 64
\]
\[
c^2 = 100
\]
Now, taking the square root of both sides:
\[
c = \sqrt{100} = 10 \, \text{ft}
\]
Thus, the length of the hypotenuse \( c \) is:
\[
\boxed{10.0} \, \text{ft}
\]
This is already to the nearest tenth.
\[
c^2 = a^2 + b^2
\]
Substituting the values:
\[
c^2 = 6^2 + 8^2
\]
Calculating the squares:
\[
c^2 = 36 + 64
\]
\[
c^2 = 100
\]
Now, taking the square root of both sides:
\[
c = \sqrt{100} = 10 \, \text{ft}
\]
Thus, the length of the hypotenuse \( c \) is:
\[
\boxed{10.0} \, \text{ft}
\]
This is already to the nearest tenth.
Answered by
GPT-4o mini
To find the length of the missing leg \( b \) in a right triangle where one leg is 12 cm and the other leg is 20 cm, we can again use the Pythagorean theorem, which states:
\[
c^2 = a^2 + b^2
\]
Here, if we consider \( c = 20 \, \text{cm} \) (the hypotenuse) and \( a = 12 \, \text{cm} \) (one leg), we can rearrange the formula to solve for \( b \):
\[
20^2 = 12^2 + b^2
\]
Calculating the squares:
\[
400 = 144 + b^2
\]
Now, subtract 144 from both sides:
\[
400 - 144 = b^2
\]
\[
256 = b^2
\]
Next, take the square root of both sides:
\[
b = \sqrt{256} = 16 \, \text{cm}
\]
Thus, the length of the missing leg \( b \) is:
\[
\boxed{16} \, \text{cm}
\]
This result is exact; rounding to the nearest tenth is not necessary in this case.
\[
c^2 = a^2 + b^2
\]
Here, if we consider \( c = 20 \, \text{cm} \) (the hypotenuse) and \( a = 12 \, \text{cm} \) (one leg), we can rearrange the formula to solve for \( b \):
\[
20^2 = 12^2 + b^2
\]
Calculating the squares:
\[
400 = 144 + b^2
\]
Now, subtract 144 from both sides:
\[
400 - 144 = b^2
\]
\[
256 = b^2
\]
Next, take the square root of both sides:
\[
b = \sqrt{256} = 16 \, \text{cm}
\]
Thus, the length of the missing leg \( b \) is:
\[
\boxed{16} \, \text{cm}
\]
This result is exact; rounding to the nearest tenth is not necessary in this case.
Answered by
GPT-4o mini
Without the specific diagram or statements that Aliyah used available to me, I can provide a general statement that is often used to explain the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)). This can be written as:
\[
c^2 = a^2 + b^2
\]
To illustrate this using a diagram, one might point out the relationship between the sides of a right triangle. For example, if there is a right triangle with legs measuring \( a \) and \( b \), and the hypotenuse measuring \( c \), Aliyah could explain that by squaring each of these lengths and summing the areas represented by \( a^2 \) and \( b^2 \), one can find the area of the square represented by \( c^2 \) (the hypotenuse).
If you have any specific options or statements that Aliyah made, please share, and I can help determine which one aligns with the explanation of the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)). This can be written as:
\[
c^2 = a^2 + b^2
\]
To illustrate this using a diagram, one might point out the relationship between the sides of a right triangle. For example, if there is a right triangle with legs measuring \( a \) and \( b \), and the hypotenuse measuring \( c \), Aliyah could explain that by squaring each of these lengths and summing the areas represented by \( a^2 \) and \( b^2 \), one can find the area of the square represented by \( c^2 \) (the hypotenuse).
If you have any specific options or statements that Aliyah made, please share, and I can help determine which one aligns with the explanation of the Pythagorean theorem.
Answered by
GPT-4o mini
To determine which statement correctly represents or explains the Pythagorean theorem, let's evaluate each one:
1. **\( 5^2 + 5^2 = 3^2 \)**
This statement is incorrect because \( 5^2 + 5^2 = 25 + 25 = 50 \) and \( 3^2 = 9 \).
2. **\( 3^2 + 4^2 = 5^2 \)**
This statement is correct. \( 3^2 + 4^2 = 9 + 16 = 25 \) and \( 5^2 = 25 \). This is a classic example of the Pythagorean theorem, where \( 3 \), \( 4 \), and \( 5 \) are the sides of a right triangle.
3. **\( 5^2 + 5^2 < 3^2 \)**
This statement is incorrect because \( 5^2 + 5^2 = 50 \) is not less than \( 3^2 = 9 \).
4. **\( 3^2 + 4^2 > 5^2 \)**
This statement is incorrect based on the previous evaluation. \( 3^2 + 4^2 = 5^2 \), so it is not greater.
Given these evaluations, the correct statement that Aliyah would use to explain the Pythagorean theorem is:
\[
\boxed{3^2 + 4^2 = 5^2}
\]
1. **\( 5^2 + 5^2 = 3^2 \)**
This statement is incorrect because \( 5^2 + 5^2 = 25 + 25 = 50 \) and \( 3^2 = 9 \).
2. **\( 3^2 + 4^2 = 5^2 \)**
This statement is correct. \( 3^2 + 4^2 = 9 + 16 = 25 \) and \( 5^2 = 25 \). This is a classic example of the Pythagorean theorem, where \( 3 \), \( 4 \), and \( 5 \) are the sides of a right triangle.
3. **\( 5^2 + 5^2 < 3^2 \)**
This statement is incorrect because \( 5^2 + 5^2 = 50 \) is not less than \( 3^2 = 9 \).
4. **\( 3^2 + 4^2 > 5^2 \)**
This statement is incorrect based on the previous evaluation. \( 3^2 + 4^2 = 5^2 \), so it is not greater.
Given these evaluations, the correct statement that Aliyah would use to explain the Pythagorean theorem is:
\[
\boxed{3^2 + 4^2 = 5^2}
\]
Answered by
GPT-4o mini
To determine which set of side lengths will form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle, the following conditions must hold:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Let's evaluate each option:
**Option 1: Side lengths 3, 3, 6**
- Check the inequalities:
1. \( 3 + 3 > 6 \) → \( 6 > 6 \) (not true)
2. \( 3 + 6 > 3 \) → \( 9 > 3 \) (true)
3. \( 3 + 6 > 3 \) → \( 9 > 3 \) (true)
Since the first condition fails, this set of lengths does not form a triangle.
---
**Option 2: Side lengths 6, 8, 10**
- Check the inequalities:
1. \( 6 + 8 > 10 \) → \( 14 > 10 \) (true)
2. \( 6 + 10 > 8 \) → \( 16 > 8 \) (true)
3. \( 8 + 10 > 6 \) → \( 18 > 6 \) (true)
All conditions are satisfied, so this set of lengths forms a triangle.
---
**Option 3: Side lengths 4, 6, 13**
- Check the inequalities:
1. \( 4 + 6 > 13 \) → \( 10 > 13 \) (not true)
2. \( 4 + 13 > 6 \) → \( 17 > 6 \) (true)
3. \( 6 + 13 > 4 \) → \( 19 > 4 \) (true)
Since the first condition fails, this set of lengths does not form a triangle.
---
Based on the evaluations:
- **Option 1**: Does not form a triangle.
- **Option 2**: Forms a triangle.
- **Option 3**: Does not form a triangle.
Thus, the response indicating which set of lengths the carpenter can use for the triangular design is:
\[
\boxed{\text{Option 2}}
\]
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Let's evaluate each option:
**Option 1: Side lengths 3, 3, 6**
- Check the inequalities:
1. \( 3 + 3 > 6 \) → \( 6 > 6 \) (not true)
2. \( 3 + 6 > 3 \) → \( 9 > 3 \) (true)
3. \( 3 + 6 > 3 \) → \( 9 > 3 \) (true)
Since the first condition fails, this set of lengths does not form a triangle.
---
**Option 2: Side lengths 6, 8, 10**
- Check the inequalities:
1. \( 6 + 8 > 10 \) → \( 14 > 10 \) (true)
2. \( 6 + 10 > 8 \) → \( 16 > 8 \) (true)
3. \( 8 + 10 > 6 \) → \( 18 > 6 \) (true)
All conditions are satisfied, so this set of lengths forms a triangle.
---
**Option 3: Side lengths 4, 6, 13**
- Check the inequalities:
1. \( 4 + 6 > 13 \) → \( 10 > 13 \) (not true)
2. \( 4 + 13 > 6 \) → \( 17 > 6 \) (true)
3. \( 6 + 13 > 4 \) → \( 19 > 4 \) (true)
Since the first condition fails, this set of lengths does not form a triangle.
---
Based on the evaluations:
- **Option 1**: Does not form a triangle.
- **Option 2**: Forms a triangle.
- **Option 3**: Does not form a triangle.
Thus, the response indicating which set of lengths the carpenter can use for the triangular design is:
\[
\boxed{\text{Option 2}}
\]
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