Find the distance between A(4, 2) and B(7, 11). The distance between the points is ___ units.
Understand the Problem
The question is asking to calculate the distance between two points A(4, 2) and B(7, 11) in a two-dimensional coordinate system. This involves using the distance formula, which is derived from the Pythagorean theorem.
Answer
The distance between the points is \( 3\sqrt{10} \) units.
Answer for screen readers
The distance between the points is ( 3\sqrt{10} ) units.
Steps to Solve
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Identify the Points The given points are ( A(4, 2) ) and ( B(7, 11) ).
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Use the Distance Formula The distance ( d ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is calculated using the formula: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
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Substitute the Values Plugging in the values from points ( A ) and ( B ): $$ d = \sqrt{(7 - 4)^2 + (11 - 2)^2} $$
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Calculate the Differences Calculate the differences inside the parentheses: $$ d = \sqrt{(3)^2 + (9)^2} $$
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Square the Differences Now, square the results: $$ d = \sqrt{9 + 81} $$
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Add and Take the Square Root Add the squared values and then take the square root: $$ d = \sqrt{90} $$
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Simplify the Square Root This can be simplified to: $$ d = 3\sqrt{10} $$
The distance between the points is ( 3\sqrt{10} ) units.
More Information
The distance formula is a direct application of the Pythagorean theorem, where the difference in ( x ) coordinates and the difference in ( y ) coordinates represent the legs of a right triangle, with the distance as the hypotenuse.
Tips
- Forgetting to square the differences in coordinates.
- Not taking the square root of the sum correctly.
- Mixing up x and y coordinates when substituting into the distance formula.
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