Work out the length of line LM. Give your answer to 3 decimal places.
Understand the Problem
The question asks for the calculation of the length of the line segment joining the points L and M on a Cartesian plane. This involves using the distance formula to find the length based on the coordinates provided.
Answer
The length of line \(LM\) is approximately $6.403$.
Answer for screen readers
The length of line (LM) is approximately $6.403$.
Steps to Solve
- Identify the coordinates of points L and M
The coordinates are given as follows:
- Point L: $(-3, 2)$
- Point M: $(1, -3)$
- Use the distance formula
The distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the formula:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
- Plug in the coordinates into the formula
Substituting the coordinates for L and M into the formula:
- For L: ( x_1 = -3, y_1 = 2 )
- For M: ( x_2 = 1, y_2 = -3 )
This gives:
$$ d = \sqrt{(1 - (-3))^2 + (-3 - 2)^2} $$
- Calculate the differences
Calculate (x_2 - x_1) and (y_2 - y_1):
- ( x_2 - x_1 = 1 + 3 = 4 )
- ( y_2 - y_1 = -3 - 2 = -5 )
- Substitute the differences back into the distance formula
Now, substitute these differences back into the formula:
$$ d = \sqrt{(4)^2 + (-5)^2} $$
- Calculate the squares and sum them
Calculating the squares:
$$ d = \sqrt{16 + 25} $$
$$ d = \sqrt{41} $$
- Calculate the square root
Now we find the square root:
$$ d \approx 6.403 $$
- Round to three decimal places
Therefore, rounding to three decimal places gives:
$$ d \approx 6.403 $$
The length of line (LM) is approximately $6.403$.
More Information
The distance formula is derived from the Pythagorean theorem, relating the lengths of the sides of a right triangle. The result tells us how far apart the two points are on the Cartesian plane.
Tips
- Not squaring the differences correctly.
- Forgetting to apply the square root at the end.
- Miscalculating the coordinates, leading to incorrect input in the formula.
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