🔑 Key Concepts
- The distance between two points on a coordinate plane can be found by forming a right triangle.
- The horizontal distance between the points becomes one leg of the triangle.
- The vertical distance between the points becomes the other leg of the triangle.
- The slanted distance between the two points is the hypotenuse.
- The distance formula is another way to write the Pythagorean theorem using coordinate differences.
✏️ Worked Example
🧠 Math Vocabulary
- Distance Formula: A formula used to find the distance between two points on the coordinate plane.
- Pythagorean Theorem: A rule for right triangles that states \(a^2+b^2=c^2\).
- Coordinate Plane: A plane formed by the \(x\)-axis and \(y\)-axis.
- Ordered Pair: A point written as \((x,y)\).
- Horizontal Distance: The difference between the \(x\)-values of two points.
- Vertical Distance: The difference between the \(y\)-values of two points.
- Hypotenuse: The longest side of a right triangle, across from the right angle.
- Pythagorean Triple: Three whole numbers that work in the Pythagorean theorem, such as \(6\), \(8\), and \(10\).
💡 Main Idea
In this lesson, students connect the Pythagorean theorem to the distance formula. The change in \(x\) gives the horizontal leg, the change in \(y\) gives the vertical leg, and the distance between the points is the hypotenuse.
📚 What You Should Already Know
Students should know how to plot ordered pairs, subtract integers, square numbers, find square roots, and use the Pythagorean theorem to find the hypotenuse of a right triangle.
🚀 What Comes Next
After connecting the distance formula to the Pythagorean theorem, students can find distances without a graph, solve coordinate plane word problems, and apply the formula to geometric figures on the coordinate plane.
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