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How to Find the Distance Between Points (-8, 5/12) and (8, 5/12)
Mathematics
Grade 11 (Senior High School)
Question Content
Find the distance between the pair of points: $\\left(-8,\\frac{5}{12}\\right)$ and $\\left(8,\\frac{5}{12}\\right)$. The distance between the pair of points is ____. (Type an integer or a simplified fraction.)
Correct Answer
16
Detailed Solution Steps
1
Step 1: Identify the coordinates of the two points. Let $(x_1, y_1) = \\left(-8,\\frac{5}{12}\\right)$ and $(x_2, y_2) = \\left(8,\\frac{5}{12}\\right)$.
2
Step 2: Notice that the $y$-coordinates of both points are equal ($\\frac{5}{12}$), so the distance between the points is the absolute difference of the $x$-coordinates. The formula for horizontal distance is $|x_2 - x_1|$.
3
Step 3: Substitute the $x$-coordinates into the formula: $|8 - (-8)| = |8 + 8| = |16| = 16$.
4
Alternative Step (using full distance formula): Use the distance formula $d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Substitute values: $d = \\sqrt{(8 - (-8))^2 + \\left(\\frac{5}{12} - \\frac{5}{12}\\right)^2} = \\sqrt{(16)^2 + 0^2} = \\sqrt{256} = 16$.
Knowledge Points Involved
1
Distance Formula for Coordinate Plane
The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. It is derived from the Pythagorean theorem, treating the line between the two points as the hypotenuse of a right triangle with legs equal to the differences in $x$ and $y$ coordinates.
2
Horizontal Distance Between Points
When two points have the same $y$-coordinate, they lie on a horizontal line. The distance between them simplifies to the absolute value of the difference of their $x$-coordinates ($|x_2 - x_1|$), since the vertical difference ($y_2 - y_1$) is 0.
3
Absolute Value
The absolute value of a number $|a|$ is its non-negative value, representing its distance from 0 on the number line. It is used in distance calculations to ensure the result is a positive, non-negative value, as distance cannot be negative.
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