Use the image to answer the question.

Question

A map shows a grid 17 units across and 12 units high, superimposed over shapes that represent streets and buildings. A key lists buildings located in Washington D.C. that correspond to points on the grid. Point A represents the White House. It is located at 4 units across from the left, and 3 units down from the top. Point B represents the Washington Monument, located at 5 units across and 9 units down. Point C represents the Natural History Museum, and is located at approximately 8 units across and 8 units down. Point D represents the Smithsonian, and is located at 10 units across and 10 units down. Point E represents the National Portrait Gallery, and is located at 12 units across and 3 units down. Point F represents the National Gallery of Art, and is located at 14 units across and 8 units down. A scale shows 200 feet and 200 meters.

Find the length between landmark A
and landmark E
.

(1 point)
units

Answer 1

To find the length between landmarks A (the White House) and E (the National Portrait Gallery), we need to determine the distance between their coordinates on the grid.

• Coordinates for Point A (White House): (4, 3)
• Coordinates for Point E (National Portrait Gallery): (12, 3)

To calculate the distance between points A and E, we can use the distance formula, which in this case simplifies to the difference in the x-coordinates since both points have the same y-coordinate:

Distance = |x2 - x1|

Where:

• x1 is the x-coordinate of Point A (4)
• x2 is the x-coordinate of Point E (12)

Distance = |12 - 4| = |8| = 8 units

Thus, the length between landmark A and landmark E is 8 units.

Answer 2

Correct.

Answer 3

Great! If you have any more questions or if there's anything else you need help with, feel free to ask!

Answer 4

Use the image to answer the question.

A map shows a grid 17 units across and 12 units high, superimposed over shapes that represent streets and buildings. A key lists buildings located in Washington D.C. that correspond to points on the grid. Point A represents the White House. It is located at 4 units across from the left, and 3 units down from the top. Point B represents the Washington Monument, located at 5 units across and 9 units down. Point C represents the Natural History Museum, and is located at approximately 8 units across and 8 units down. Point D represents the Smithsonian, and is located at 10 units across and 10 units down. Point E represents the National Portrait Gallery, and is located at 12 units across and 3 units down. Point F represents the National Gallery of Art, and is located at 14 units across and 8 units down. A scale shows 200 feet and 200 meters.

Find the length between landmark B
and F
. Round the answer to the nearest hundredth, if necessary.
units
Question 3: Use the image to answer the question.

A graph shows four quadrants with the x-axis and y-axis both ranging from negative 10 to positive 10 in increments of 1. Unlabeled points are plotted at left parenthesis negative 6 comma 8 right parenthesis, left parenthesis 4 comma 8 right parenthesis, left parenthesis 4 comma negative 3 right parenthesis, and left parenthesis negative 6 comma negative 3 right parenthesis. A straight line joins the four points forming a rectangle.

What is the length of the diagonal of the rectangle? Round your answer to the nearest hundredth, if necessary.
units
Question 4: Find the distance between (2, 3)
and (−4, −9)
. Round your answer to the nearest hundredth, if necessary.(1 point)
units
Question 5: Find the distance between (−8, −3)
and (−2, −5)
. Round your answer to the nearest hundredth, if necessary.(1 point)
units

Answer 5

Let’s solve each question step by step:

Question 1: Distance between B and F

• Coordinates:
  • Point B (Washington Monument): (5, 9)
  • Point F (National Gallery of Art): (14, 8)

We can use the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates: \[ = \sqrt{(14 - 5)^2 + (8 - 9)^2} \] \[ = \sqrt{(9)^2 + (-1)^2} \] \[ = \sqrt{81 + 1} \] \[ = \sqrt{82} \] Calculating this gives: \[ \approx 9.06 \text{ units (rounded to the nearest hundredth)} \]

Question 2: Length of the diagonal of the rectangle

• Coordinates for the rectangle:
  • Point 1: (-6, 8)
  • Point 2: (4, 8)
  • Point 3: (4, -3)
  • Point 4: (-6, -3)

The diagonal we can consider is from Point 1 (-6, 8) to Point 3 (4, -3).

Using the distance formula: \[ \text{Distance} = \sqrt{(4 - (-6))^2 + (-3 - 8)^2} \] \[ = \sqrt{(4 + 6)^2 + (-11)^2} \] \[ = \sqrt{(10)^2 + (-11)^2} \] \[ = \sqrt{100 + 121} \] \[ = \sqrt{221} \] Calculating this gives: \[ \approx 14.83 \text{ units (rounded to the nearest hundredth)} \]

Question 3: Distance between (2, 3) and (−4, −9)

Using the distance formula: \[ \text{Distance} = \sqrt{(-4 - 2)^2 + (-9 - 3)^2} \] \[ = \sqrt{(-6)^2 + (-12)^2} \] \[ = \sqrt{36 + 144} \] \[ = \sqrt{180} \] Calculating this gives: \[ \approx 13.42 \text{ units (rounded to the nearest hundredth)} \]

Question 4: Distance between (−8, −3) and (−2, −5)

Using the distance formula: \[ \text{Distance} = \sqrt{(-2 - (-8))^2 + (-5 - (-3))^2} \] \[ = \sqrt{(-2 + 8)^2 + (-5 + 3)^2} \] \[ = \sqrt{(6)^2 + (-2)^2} \] \[ = \sqrt{36 + 4} \] \[ = \sqrt{40} \] Calculating this gives: \[ \approx 6.32 \text{ units (rounded to the nearest hundredth)} \]

Summary of Answers:

1. Distance between B and F: 9.06 units
2. Length of the diagonal of the rectangle: 14.83 units
3. Distance between (2, 3) and (−4, −9): 13.42 units
4. Distance between (−8, −3) and (−2, −5): 6.32 units

Feel free to ask if you have more questions!

Answer 6

14.83 is wrong....

Answer 7

Let's carefully recalculate the length of the diagonal of the rectangle from the coordinates given.

Coordinates for the rectangle:

• Point 1: (-6, 8)
• Point 2: (4, 8)
• Point 3: (4, -3)
• Point 4: (-6, -3)

To find the diagonal length, you can use either of the diagonals. Let's find the diagonal from Point 1 (-6, 8) to Point 3 (4, -3).

Using the distance formula again: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates: \[ \text{Distance} = \sqrt{(4 - (-6))^2 + (-3 - 8)^2} \] \[ = \sqrt{(4 + 6)^2 + (-11)^2} \] \[ = \sqrt{(10)^2 + (-11)^2} \] \[ = \sqrt{100 + 121} \] \[ = \sqrt{221} \]

Calculating \(\sqrt{221}\): \[ \sqrt{221} \approx 14.83 \]

Since you've confirmed that my previous answer was incorrect, let's double-check the choices of diagonal points.

Calculating the diagonal from Point 2 (4, 8) to Point 4 (-6, -3): \[ \text{Distance} = \sqrt{(-6 - 4)^2 + (-3 - 8)^2} \] \[ = \sqrt{(-10)^2 + (-11)^2} \] \[ = \sqrt{100 + 121} \] \[ = \sqrt{221} \approx 14.83 \]

Upon reviewing, the calculations still yield \(\sqrt{221} \approx 14.83\).

Could you please clarify how you determined that this result was incorrect? If there are specific points or details I should know, I’d be happy to help resolve it!