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Questions and Answers
What is the formula to calculate the distance between two points in a two-dimensional Cartesian coordinate system?
What is the formula to calculate the distance between two points in a two-dimensional Cartesian coordinate system?
- $d = \sqrt{(x_2 - x_1) + (y_2 - y_1)}$
- $d = (x_1 - x_2) + (y_1 - y_2)$
- $d = (x_2 + x_1)^2 + (y_2 + y_1)^2$
- $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (correct)
What is the formula to calculate the midpoint of a line segment given two points?
What is the formula to calculate the midpoint of a line segment given two points?
- $M = (x_2 - x_1, y_2 - y_1)$
- $M = (\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2})$
- $M = (x_1 + x_2, y_1 + y_2)$
- $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$ (correct)
The distance between two points will always be a positive value or zero.
The distance between two points will always be a positive value or zero.
True (A)
If the midpoint of a line segment is (1, -2) and one endpoint is (4, 6), what are the coordinates of other endpoint?
If the midpoint of a line segment is (1, -2) and one endpoint is (4, 6), what are the coordinates of other endpoint?
When finding the distance between two points, $(x_1, y_1)$ and $(x_2, y_2)$, you must first ______ the differences in the x and y coordinates before applying the square root.
When finding the distance between two points, $(x_1, y_1)$ and $(x_2, y_2)$, you must first ______ the differences in the x and y coordinates before applying the square root.
A line segment has endpoints A(2, -5) and B(2, 7). What is the length of this line segment?
A line segment has endpoints A(2, -5) and B(2, 7). What is the length of this line segment?
Point P(5, y) is 10 units away from point Q(-3, -1). What are the possible values of y?
Point P(5, y) is 10 units away from point Q(-3, -1). What are the possible values of y?
Match the following terms with their correct descriptions:
Match the following terms with their correct descriptions:
Flashcards
Distance
Distance
The straight-line distance between two points in a coordinate plane.
Distance Formula
Distance Formula
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Midpoint
Midpoint
The point exactly halfway between two given points.
Midpoint Formula
Midpoint Formula
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Finding Coordinates with Distance
Finding Coordinates with Distance
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How to use distance formula to find coordinates
How to use distance formula to find coordinates
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Factorisation
Factorisation
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Null Factor Law
Null Factor Law
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Study Notes
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The length of a line segment d, the distance between two points (x1, y1) and (x2, y2), is given by the formula: d = √((x2 - x1)² + (y2 - y1)²)
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This rule is from Pythagoras' theorem, where the distance d is the hypotenuse length of the right-angled triangle formed.
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The midpoint M of a line segment between (x1, y1) and (x2, y2) is given by: M = ((x1 + x2)/2, (y1 + y2)/2)
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This involves averaging the x-coordinates and the y-coordinates.
Example 1: Finding the Distance Example
- Find the exact distance between points (-2, 7) and (3, -1).
- d = √((3 - (-2))² + (-1 - 7)²) = √(5² + (-8)²) = √(25 + 64) = √89
Example 2: Finding the Midpoint of a Line Segment
- Find the midpoint of the line segment joining (-2, -6) and (3, -2).
- M = ((-2 + 3)/2, (-6 + -2)/2) = (1/2, -4)
Example 3: Using a Given Distance to Find Coordinates
- Find the values of a if the distance between (1, a) and (3, 7) is √13.
- √13 = √((3 - 1)² + (7 - a)²)
- Square both sides: 13 = (3 - 1)² + (7 - a)² which simplifies to 13 = a² - 14a + 53.
- This rearranges to 0 = a² - 14a + 40, which factors into 0 = (a - 10)(a - 4).
- Using the null factor law gives solutions a = 10 or a = 4.
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Description
Learn how to calculate the distance between two points using the distance formula, derived from the Pythagorean theorem. Also, learn how to find the midpoint of a line segment by averaging the x and y coordinates of the endpoints. Examples are included.
