Finding The Length Of A Hypotenuse In Simplest Radical Form

🔑 Key Concepts

• The hypotenuse is the longest side of a right triangle and is across from the right angle.
• To find the hypotenuse, add the squares of the two legs.
• After adding the squares, take the square root to find the side length.
• If the square root is not a perfect square, write the answer in simplest radical form.
• To simplify a radical, look for the largest perfect-square factor inside the radical.

✏️ Worked Examples

🧠 Math Vocabulary

• Pythagorean Theorem: A rule for right triangles that states \(a^2+b^2=c^2\).
• Right Triangle: A triangle with one \(90^\circ\) angle.
• Hypotenuse: The longest side of a right triangle, located across from the right angle.
• Legs: The two sides of a right triangle that form the right angle.
• Perfect Square: A number made by multiplying a whole number by itself, such as 25, 36, 49, or 64.
• Square Root: A number that is multiplied by itself to make another number.
• Radical: A square root expression, such as \(\sqrt{80}\).
• Simplest Radical Form: A radical expression with no perfect-square factor left inside the square root.

💡 Main Idea

In this lesson, students find the hypotenuse of right triangles using the Pythagorean theorem and express answers in simplest radical form when the result is not a perfect square.

📚 What You Should Already Know

Students should know how to square numbers, find square roots of perfect squares, identify the hypotenuse of a right triangle, and simplify basic radical expressions.

🚀 What Comes Next

After finding hypotenuse lengths, students can use the Pythagorean theorem to find missing legs, solve word problems, find diagonal distances on the coordinate plane, and apply right triangle reasoning in geometry problems.

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