Question
Distance and the Pythagorean Theorem Quick Check
1. Use the image to answer the question.
A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 5. Data points marked are point A at left parenthesis 1 comma 2 right parenthesis, point B at left parenthesis 3 comma 2 right parenthesis, point C at left parenthesis 3 comma 4 right parenthesis, and point D at left parenthesis 1 comma 4 right parenthesis. A line joins points A to B, B to C, C to D, and D to A, forming a rectangle.
Apply the Pythagorean Theorem to find the length of the diagonal of the square. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
4
4
4.24
4.24
8
8
2.83
2. Apply the Pythagorean Theorem to find the length between (2, 5) and (7, 3) . Round your answer to the nearest hundredth, if necessary.(1 point)
Responses
29
29
2.65
2.65
4.58
4.58
5.39
3. Apply the Pythagorean Theorem to find the length between (−1, 6) and (5, −2) . Round your answer to the nearest hundredth, if necessary.(1 point)
Responses
9.22
9.22
10
10
100
100
14
4. Use the image to answer the question.
A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Unlabeled data points marked are at left parenthesis negative 2 comma negative 1 right parenthesis, and at left parenthesis 2 comma 4 right parenthesis. A straight line joins the two lines with a positive slope.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
41
41
9
9
6:40
6:40
5.57
5.Use the image to answer the question.
A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Data points marked are point A at left parenthesis 2 comma negative 3 right parenthesis, and point B at left parenthesis 5 comma 5 right parenthesis.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
73
73
8.54
8.54
7.62
7.62
11
1. Use the image to answer the question.
A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 5. Data points marked are point A at left parenthesis 1 comma 2 right parenthesis, point B at left parenthesis 3 comma 2 right parenthesis, point C at left parenthesis 3 comma 4 right parenthesis, and point D at left parenthesis 1 comma 4 right parenthesis. A line joins points A to B, B to C, C to D, and D to A, forming a rectangle.
Apply the Pythagorean Theorem to find the length of the diagonal of the square. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
4
4
4.24
4.24
8
8
2.83
2. Apply the Pythagorean Theorem to find the length between (2, 5) and (7, 3) . Round your answer to the nearest hundredth, if necessary.(1 point)
Responses
29
29
2.65
2.65
4.58
4.58
5.39
3. Apply the Pythagorean Theorem to find the length between (−1, 6) and (5, −2) . Round your answer to the nearest hundredth, if necessary.(1 point)
Responses
9.22
9.22
10
10
100
100
14
4. Use the image to answer the question.
A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Unlabeled data points marked are at left parenthesis negative 2 comma negative 1 right parenthesis, and at left parenthesis 2 comma 4 right parenthesis. A straight line joins the two lines with a positive slope.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
41
41
9
9
6:40
6:40
5.57
5.Use the image to answer the question.
A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Data points marked are point A at left parenthesis 2 comma negative 3 right parenthesis, and point B at left parenthesis 5 comma 5 right parenthesis.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
73
73
8.54
8.54
7.62
7.62
11
Answers
Answer
u sure?
Answered by
GPT-4o
Let's solve each problem step by step using the Pythagorean Theorem.
1. Apply the Pythagorean Theorem to find the length of the diagonal of the rectangle.
Points A (1, 2) and B (3, 2) have a horizontal distance of 2 units. Points B (3, 2) and C (3, 4) have a vertical distance of 2 units.
Since these distances are the legs of a right triangle, the diagonal is the hypotenuse.
\[
\text{Length} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
The correct answer is:
```
2.83
```
2. Apply the Pythagorean Theorem to find the length between (2, 5) and (7, 3).
The horizontal distance \( \Delta x = 7 - 2 = 5 \)
The vertical distance \( \Delta y = 5 - 3 = 2 \)
The distance is:
\[
\sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.39
\]
The correct answer is:
```
5.39
```
3. Apply the Pythagorean Theorem to find the length between (-1, 6) and (5, -2).
The horizontal distance \( \Delta x = 5 - (-1) = 6 \)
The vertical distance \( \Delta y = 6 - (-2) = 8 \)
The distance is:
\[
\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
The correct answer is:
```
10
```
4. Apply the Pythagorean Theorem to find the length between (-2, -1) and (2, 4).
The horizontal distance \( \Delta x = 2 - (-2) = 4 \)
The vertical distance \( \Delta y = 4 - (-1) = 5 \)
The distance is:
\[
\sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40
\]
The correct answer is:
```
6.40
```
5. Apply the Pythagorean Theorem to find the length between (2, -3) and (5, 5).
The horizontal distance \( \Delta x = 5 - 2 = 3 \)
The vertical distance \( \Delta y = 5 - (-3) = 8 \)
The distance is:
\[
\sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54
\]
The correct answer is:
```
8.54
```
1. Apply the Pythagorean Theorem to find the length of the diagonal of the rectangle.
Points A (1, 2) and B (3, 2) have a horizontal distance of 2 units. Points B (3, 2) and C (3, 4) have a vertical distance of 2 units.
Since these distances are the legs of a right triangle, the diagonal is the hypotenuse.
\[
\text{Length} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
The correct answer is:
```
2.83
```
2. Apply the Pythagorean Theorem to find the length between (2, 5) and (7, 3).
The horizontal distance \( \Delta x = 7 - 2 = 5 \)
The vertical distance \( \Delta y = 5 - 3 = 2 \)
The distance is:
\[
\sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.39
\]
The correct answer is:
```
5.39
```
3. Apply the Pythagorean Theorem to find the length between (-1, 6) and (5, -2).
The horizontal distance \( \Delta x = 5 - (-1) = 6 \)
The vertical distance \( \Delta y = 6 - (-2) = 8 \)
The distance is:
\[
\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
The correct answer is:
```
10
```
4. Apply the Pythagorean Theorem to find the length between (-2, -1) and (2, 4).
The horizontal distance \( \Delta x = 2 - (-2) = 4 \)
The vertical distance \( \Delta y = 4 - (-1) = 5 \)
The distance is:
\[
\sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40
\]
The correct answer is:
```
6.40
```
5. Apply the Pythagorean Theorem to find the length between (2, -3) and (5, 5).
The horizontal distance \( \Delta x = 5 - 2 = 3 \)
The vertical distance \( \Delta y = 5 - (-3) = 8 \)
The distance is:
\[
\sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54
\]
The correct answer is:
```
8.54
```
Answered by
GPT-4o
Let's go through each problem again to ensure we have the correct calculations.
1. **Length of the diagonal of the rectangle:**
Points are:
- A: (1, 2)
- B: (3, 2)
- C: (3, 4)
- D: (1, 4)
The horizontal distance between A and B or C and D is \( \Delta x = 3 - 1 = 2 \). The vertical distance between B and C or A and D is \( \Delta y = 4 - 2 = 2 \).
Diagonal calculation:
\[
\text{Diagonal} = \sqrt{(3 - 1)^2 + (4 - 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
2. **Length between (2, 5) and (7, 3):**
Differences:
\[
\Delta x = 7 - 2 = 5
\]
\[
\Delta y = 3 - 5 = -2
\]
Distance calculation:
\[
\sqrt{(7 - 2)^2 + (3 - 5)^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.39
\]
3. **Length between (-1, 6) and (5, -2):**
Differences:
\[
\Delta x = 5 - (-1) = 6
\]
\[
\Delta y = -2 - 6 = -8
\]
Distance calculation:
\[
\sqrt{(5 - (-1))^2 + (-2 - 6)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
4. **Length between (-2, -1) and (2, 4):**
Differences:
\[
\Delta x = 2 - (-2) = 4
\]
\[
\Delta y = 4 - (-1) = 5
\]
Distance calculation:
\[
\sqrt{(2 - (-2))^2 + (4 - (-1))^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40
\]
5. **Length between (2, -3) and (5, 5):**
Differences:
\[
\Delta x = 5 - 2 = 3
\]
\[
\Delta y = 5 - (-3) = 8
\]
Distance calculation:
\[
\sqrt{(5 - 2)^2 + (5 - (-3))^2} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54
\]
So, the answers are:
- **2.83**
- **5.39**
- **10**
- **6.40**
- **8.54**
These all check out based on the re-evaluation of the given problems.
1. **Length of the diagonal of the rectangle:**
Points are:
- A: (1, 2)
- B: (3, 2)
- C: (3, 4)
- D: (1, 4)
The horizontal distance between A and B or C and D is \( \Delta x = 3 - 1 = 2 \). The vertical distance between B and C or A and D is \( \Delta y = 4 - 2 = 2 \).
Diagonal calculation:
\[
\text{Diagonal} = \sqrt{(3 - 1)^2 + (4 - 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
2. **Length between (2, 5) and (7, 3):**
Differences:
\[
\Delta x = 7 - 2 = 5
\]
\[
\Delta y = 3 - 5 = -2
\]
Distance calculation:
\[
\sqrt{(7 - 2)^2 + (3 - 5)^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.39
\]
3. **Length between (-1, 6) and (5, -2):**
Differences:
\[
\Delta x = 5 - (-1) = 6
\]
\[
\Delta y = -2 - 6 = -8
\]
Distance calculation:
\[
\sqrt{(5 - (-1))^2 + (-2 - 6)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
4. **Length between (-2, -1) and (2, 4):**
Differences:
\[
\Delta x = 2 - (-2) = 4
\]
\[
\Delta y = 4 - (-1) = 5
\]
Distance calculation:
\[
\sqrt{(2 - (-2))^2 + (4 - (-1))^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40
\]
5. **Length between (2, -3) and (5, 5):**
Differences:
\[
\Delta x = 5 - 2 = 3
\]
\[
\Delta y = 5 - (-3) = 8
\]
Distance calculation:
\[
\sqrt{(5 - 2)^2 + (5 - (-3))^2} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54
\]
So, the answers are:
- **2.83**
- **5.39**
- **10**
- **6.40**
- **8.54**
These all check out based on the re-evaluation of the given problems.
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