Find the Distance Between Two Points

🔑 Key Concepts

  • The distance between two points on a coordinate plane can be found by creating a right triangle.
  • The horizontal change between \(A(-4,-10)\) and \(B(8,6)\) is \(12\) units.
  • The vertical change between the points is \(16\) units.
  • The distance between the points is the hypotenuse of the right triangle.

💡 Main Idea

To find the distance between \(A(-4,-10)\) and \(B(8,6)\), students can use the Pythagorean theorem by drawing a right triangle or use the distance formula directly. Both methods use the same horizontal and vertical changes and lead to the same distance.

✏️ Worked Example

🧠 Math Vocabulary

  • Coordinate Plane: A grid formed by the \(x\)-axis and \(y\)-axis.
  • Ordered Pair: A pair of numbers written as \((x,y)\) that shows the location of a point.
  • Horizontal Change: The left-to-right distance between two points, found by comparing the \(x\)-coordinates.
  • Vertical Change: The up-and-down distance between two points, found by comparing the \(y\)-coordinates.
  • Hypotenuse: The longest side of a right triangle, across from the right angle.
  • 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 \(a^2+b^2=c^2\).
  • Pythagorean Triple: A set of three whole numbers that satisfies \(a^2+b^2=c^2\).

📚 What You Should Already Know

Students should know how to read ordered pairs, find horizontal and vertical distance on a coordinate plane, square numbers, take square roots, and apply the Pythagorean theorem to right triangles.

🚀 What Comes Next

After finding distance using a right triangle, students can apply the distance formula directly to ordered pairs and solve coordinate geometry problems without needing to draw every triangle.

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