Distance and the Pythagorean Theorem Quick Check
1 of 51 of 5 Items
Question
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
8
8
2.83
2.83
4.24
4.24
4
Distance and the Pythagorean Theorem Quick Check
2 of 52 of 5 Items
Question
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
5.39
5.39
2.65
2.65
4.58
4.58
29
29
Distance and the Pythagorean Theorem Quick Check
3 of 53 of 5 Items
Question
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
14
14
10
10
9.22
9.22
100
Distance and the Pythagorean Theorem Quick Check
4 of 54 of 5 Items
Question
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
5.57
5.57
9
9
6:40
Distance and the Pythagorean Theorem Quick Check
5 of 55 of 5 Items
Question
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
8.54
8.54
7.62
7.62
11
11
73
73
Certainly! Let's go through each question once again to triple-check the calculations:
1. Applying the Pythagorean theorem to find the length of the diagonal of the square formed by the points A(1, 2), B(3, 2), C(3, 4), and D(1, 4):
The length of AB is 3 - 1 = 2 (horizontal distance)
The length of BC is 4 - 2 = 2 (vertical distance)
Using the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the sides, and c is the length of the hypotenuse (diagonal in this case).
a^2 + b^2 = c^2
2^2 + 2^2 = c^2
4 + 4 = c^2
8 = c^2
c = √8 ≈ 2.83
The length of the diagonal is approximately 2.83.
2. Applying the Pythagorean theorem to find the length between (2, 5) and (7, 3):
The length of the horizontal distance is 7 - 2 = 5
The length of the vertical distance is 5 - 3 = 2
Using the Pythagorean theorem:
a^2 + b^2 = c^2
5^2 + 2^2 = c^2
25 + 4 = c^2
29 = c^2
c = √29 ≈ 5.39
The length between the two points is approximately 5.39.
3. Applying the Pythagorean theorem to find the length between (-1, 6) and (5, -2):
The length of the horizontal distance is 5 - (-1) = 6
The length of the vertical distance is -2 - 6 = -8 (we consider the absolute value)
Using the Pythagorean theorem:
a^2 + b^2 = c^2
6^2 + (-8)^2 = c^2
36 + 64 = c^2
100 = c^2
c = √100 = 10
The length between the two points is 10.
4. Applying the Pythagorean theorem to find the length between (-2, -1) and (2, 4):
The length of the horizontal distance is 2 - (-2) = 4
The length of the vertical distance is 4 - (-1) = 5
Using the Pythagorean theorem:
a^2 + b^2 = c^2
4^2 + 5^2 = c^2
16 + 25 = c^2
41 = c^2
c = √41 ≈ 6.40
The length between the two points is approximately 6.40.
5. Applying the Pythagorean theorem to find the length between (2, -3) and (5, 5):
The length of the horizontal distance is 5 - 2 = 3
The length of the vertical distance is 5 - (-3) = 8
Using the Pythagorean theorem:
a^2 + b^2 = c^2
3^2 + 8^2 = c^2
9 + 64 = c^2
73 = c^2
c = √73 ≈ 8.54
The length between the two points is approximately 8.54.
Upon triple-checking, the calculations remain the same.
To find the length between two points using the Pythagorean Theorem, we need to calculate the distance between the x-coordinates and the distance between the y-coordinates. Then, we can use these distances to find the hypotenuse, which represents the distance between the two points.
Let's go through each question and calculate the length using this method:
1. In the first question, we are given the points A(1, 2), B(3, 2), C(3, 4), and D(1, 4). These points form a rectangle. To find the length of the diagonal (which is also the hypotenuse of the rectangle), we can use the distance formula (d^2 = (x2 - x1)^2 + (y2 - y1)^2) or the Pythagorean Theorem (c^2 = a^2 + b^2).
Using the distance formula:
Length of diagonal = √[(3 - 1)^2 + (4 - 2)^2]
= √[(2)^2 + (2)^2]
= √[4 + 4]
= √8
≈ 2.83
So the answer to question 1 is 2.83.
2. In the second question, we are given the points (2, 5) and (7, 3). To find the distance between these points, we can again use the distance formula or the Pythagorean Theorem.
Using the distance formula:
Distance = √[(7 - 2)^2 + (3 - 5)^2]
= √[(5)^2 + (-2)^2]
= √[25 + 4]
= √29
≈ 5.39
So the answer to question 2 is 5.39.
3. In the third question, we are given the points (-1, 6) and (5, -2). Again, we can use the distance formula or the Pythagorean Theorem.
Using the distance formula:
Distance = √[(5 - (-1))^2 + (-2 - 6)^2]
= √[(5 + 1)^2 + (-8)^2]
= √[6^2 + 64]
= √[36 + 64]
= √100
= 10
So the answer to question 3 is 10.
4. In the fourth question, we are given the points (-2, -1) and (2, 4). Again, we can use the distance formula or the Pythagorean Theorem.
Using the distance formula:
Distance = √[(2 - (-2))^2 + (4 - (-1))^2]
= √[(2 + 2)^2 + (4 + 1)^2]
= √[4^2 + 5^2]
= √[16 + 25]
= √41
≈ 6.40
So the answer to question 4 is 6.40.
5. In the fifth question, we are given the points (2, -3) and (5, 5). Again, we can use the distance formula or the Pythagorean Theorem.
Using the distance formula:
Distance = √[(5 - 2)^2 + (5 - (-3))^2]
= √[(3)^2 + (5 + 3)^2]
= √[9 + 64]
= √73
≈ 8.54
So the answer to question 5 is 8.54.
Therefore, the correct answers to the questions are:
1. 2.83
2. 5.39
3. 10
4. 6.40
5. 8.54
can you show the math and double check everything
Certainly! Let's go through each question and calculate the answers:
1. Applying the Pythagorean theorem to find the length of the diagonal of the square formed by the points A(1, 2), B(3, 2), C(3, 4), and D(1, 4):
The length of AB is 3 - 1 = 2 (horizontal distance)
The length of BC is 4 - 2 = 2 (vertical distance)
Using the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the sides, and c is the length of the hypotenuse (diagonal in this case).
a^2 + b^2 = c^2
2^2 + 2^2 = c^2
4 + 4 = c^2
8 = c^2
c = √8 ≈ 2.83
The length of the diagonal is approximately 2.83.
2. Applying the Pythagorean theorem to find the length between (2, 5) and (7, 3):
The length of the horizontal distance is 7 - 2 = 5
The length of the vertical distance is 5 - 3 = 2
Using the Pythagorean theorem:
a^2 + b^2 = c^2
5^2 + 2^2 = c^2
25 + 4 = c^2
29 = c^2
c = √29 ≈ 5.39
The length between the two points is approximately 5.39.
3. Applying the Pythagorean theorem to find the length between (-1, 6) and (5, -2):
The length of the horizontal distance is 5 - (-1) = 6
The length of the vertical distance is -2 - 6 = -8 (we consider the absolute value)
Using the Pythagorean theorem:
a^2 + b^2 = c^2
6^2 + (-8)^2 = c^2
36 + 64 = c^2
100 = c^2
c = √100 = 10
The length between the two points is 10.
4. Applying the Pythagorean theorem to find the length between (-2, -1) and (2, 4):
The length of the horizontal distance is 2 - (-2) = 4
The length of the vertical distance is 4 - (-1) = 5
Using the Pythagorean theorem:
a^2 + b^2 = c^2
4^2 + 5^2 = c^2
16 + 25 = c^2
41 = c^2
c = √41 ≈ 6.40
The length between the two points is approximately 6.40.
5. Applying the Pythagorean theorem to find the length between (2, -3) and (5, 5):
The length of the horizontal distance is 5 - 2 = 3
The length of the vertical distance is 5 - (-3) = 8
Using the Pythagorean theorem:
a^2 + b^2 = c^2
3^2 + 8^2 = c^2
9 + 64 = c^2
73 = c^2
c = √73 ≈ 8.54
The length between the two points is approximately 8.54.
I double-checked all the calculations and they should be correct.
can you triple check
Question Use the image to answer the question. 25 3, 26 5, 27 4, 29 0, 30 5. What number represents the gap in the dot plot? (1 point)
Answer is 29.
The correct answers are:
1. 4.24
2. 5.39
3. 9.22
4. 5.57
5. 8.54