Distance and the Pythagorean Theorem Practice

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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.

(1 point)
units

Answer 1

To find the distance between points B and F using the Pythagorean theorem, we first determine the coordinates of points B and F:

• Point B: (5, 9)
• Point F: (14, 8)

Next, we can use the formula for the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

\[ d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2} \]

Substituting in the coordinates for B and F:

• \( x_1 = 5 \)
• \( y_1 = 9 \)
• \( x_2 = 14 \)
• \( y_2 = 8 \)

Calculating the differences:

\[ d = \sqrt{{(14 - 5)}^2 + {(8 - 9)}^2} \] \[ = \sqrt{{(9)}^2 + {(-1)}^2} \] \[ = \sqrt{81 + 1} \] \[ = \sqrt{82} \]

Now calculate \( \sqrt{82} \):

\[ \sqrt{82} \approx 9.055 \]

Rounding this to the nearest hundredth:

\[ d \approx 9.06 \]

Thus, the length between landmark B and F is approximately 9.06 units.