📚 Practice With Worksheets
Practice using the Pythagorean Theorem and distance formula ideas to find missing side lengths, diagonal distances, and exact or rounded answers.
▶️ Learn With Videos
Watch these lessons to review how the distance formula and Pythagorean Theorem are used to find exact and approximate distances between two points.
🛠️ How to Use This Tool
Click New Problem to generate two points on the coordinate plane.
Use the connected segment to find the distance between the points. You may enter a decimal answer rounded to the nearest tenth or an exact answer in simplest radical form.
Use Show Right Triangle to see the horizontal and vertical changes that connect the distance formula to the Pythagorean Theorem.
🧠 Things to Notice
The distance between two points is the length of the diagonal segment connecting them.
The horizontal change and vertical change form the legs of a right triangle.
The distance is the hypotenuse, so the distance formula is really the Pythagorean Theorem on the coordinate plane.
📘 Vocabulary
Distance Formula: A formula used to find the distance between two points on the coordinate plane.
Coordinate Plane: A graph formed by the x-axis and y-axis where points are located using ordered pairs.
Ordered Pair: A pair of numbers written as \((x,y)\) that tells the location of a point.
x-coordinate: The first number in an ordered pair that tells the horizontal position.
y-coordinate: The second number in an ordered pair that tells the vertical position.
Horizontal Distance: The left-or-right change between two points, often called \(\Delta x\).
Vertical Distance: The up-or-down change between two points, often called \(\Delta y\).
Right Triangle: A triangle with one 90-degree angle.
Legs: The two sides of a right triangle that form the right angle.
Hypotenuse: The longest side of a right triangle, opposite the right angle.
Pythagorean Theorem: The relationship \(a^2+b^2=c^2\) for the side lengths of a right triangle.
Radical Form: An exact answer written using a square root, such as \(2\sqrt{5}\).
Nearest Tenth: A decimal rounded to one digit after the decimal point.
Approximate Distance: A rounded decimal value used when the exact distance is irrational.
📐 Distance Formula Rules
To find the distance between two points, subtract the x-coordinates and y-coordinates, square each difference, add them, and take the square root.
\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
The same idea comes from the Pythagorean Theorem, where the horizontal and vertical changes are the legs of a right triangle.
\(a^2+b^2=c^2\)
If the square root is not a perfect square, the distance may be written as a rounded decimal or in simplest radical form.
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