Distance Between Two Points in Simplest Radical Form

💡 Main Idea

In this short lesson, students find the distance between two points on a coordinate plane by creating a right triangle. The horizontal and vertical distances become the legs, and the segment \(\overline{AB}\) becomes the hypotenuse. The final answer is written in simplest radical form.

✏️ Worked Example

Problem from the video: Find the distance of \(\overline{AB}\). Express your answer in simplest radical form.

The segment from \(A\) to \(B\) is the slanted distance, so it is the hypotenuse of a right triangle.

The vertical distance is \(15\) units.

The horizontal distance is \(10\) units.

\(a^2+b^2=c^2\)

\(15^2+10^2=c^2\)

\(225+100=c^2\)

\(325=c^2\)

\(\sqrt{325}=c\)

Now simplify the radical.

\(\sqrt{325}=\sqrt{25\cdot13}\)

\(\sqrt{325}=5\sqrt{13}\)

Answer: The distance of \(\overline{AB}\) is \(\boxed{5\sqrt{13}}\) units.

Why This Works

The distance formula and the Pythagorean theorem are closely connected. On the graph, 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.

🧠 Math Vocabulary

• Coordinate Plane: A grid formed by the \(x\)-axis and \(y\)-axis.
• Distance: The length between two points.
• Horizontal Distance: The left-to-right distance between two points.
• Vertical Distance: The up-and-down distance between two points.
• Hypotenuse: The longest side of a right triangle, across from the right angle.
• Pythagorean Theorem: A rule for right triangles that states \(a^2+b^2=c^2\).
• Simplest Radical Form: A square root expression with no perfect-square factor left inside the radical.

📚 What You Should Already Know

Students should know how to read points on a coordinate plane, count horizontal and vertical distance, identify the hypotenuse of a right triangle, square numbers, and simplify square roots.

🚀 What Comes Next

After using a graph to find distance, students can use the distance formula directly from ordered pairs and apply the same idea to geometry problems on the coordinate plane.

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