Distance and the Pythagorean Theorem Practice
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Question
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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
To find the length between landmark B and F, we can use the Pythagorean Theorem.
The horizontal distance between B and F is 14 - 5 = 9 units.
The vertical distance between B and F is 8 - 9 = -1 units.
So, we have a right triangle with horizontal distance 9 units and vertical distance -1 units.
Using the Pythagorean theorem, the length between B and F is:
√(9^2 + (-1)^2) = √(81 + 1) = √82 ≈ 9.06 units
Therefore, the length between landmark B and F is approximately 9.06 units.
To find the length between landmark B and F, we can use the distance formula, which is derived from the Pythagorean theorem in a Cartesian coordinate system. The formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Where (x1, y1) and (x2, y2) are the coordinates of the two points.
In this case, B has coordinates (5, 9) and F has coordinates (14, 8). Substituting these values into the formula, we get:
d = √((14 - 5)^2 + (8 - 9)^2)
= √(9^2 + (-1)^2)
= √(81 + 1)
= √82
≈ 9.06
So, the length between landmark B and F is approximately 9.06 units.
To find the distance between point B (representing the Washington Monument) and point F (representing the National Gallery of Art), we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
First, we need to find the horizontal and vertical distances (or legs) between these two points. From the information given, we know that point B is located at 5 units across and 9 units down, while point F is located at 14 units across and 8 units down.
To find the horizontal distance between B and F, we subtract the x-coordinates: 14 - 5 = 9 units.
To find the vertical distance between B and F, we subtract the y-coordinates: 8 - 9 = -1 units.
Now, we can use these distances to calculate the length between B and F using the Pythagorean theorem.
The square of the hypotenuse (d) is equal to the sum of the squares of the legs.
d^2 = (horizontal distance)^2 + (vertical distance)^2
d^2 = 9^2 + (-1)^2
d^2 = 81 + 1
d^2 = 82
Therefore, the square of the length between B and F is 82 units squared.
Finally, to find the actual length between B and F, we need to take the square root of 82:
d = √82
d ≈ 9.06 units.
Therefore, the length between landmark B and F is approximately 9.06 units.