Pythagorean Theorem in Simplest Radical Form

🔑 Key Concepts

• The Pythagorean theorem can be used to find a missing leg or hypotenuse.
• Some answers contain square roots that must be simplified.
• To simplify a radical, look for the largest perfect square factor.
• Checking your answer by substituting it back into the original equation helps verify your work.
• Always identify whether the missing side is a leg or the hypotenuse before solving.

✏️ Worked Examples

Example 1: Missing Leg in Simplest Radical Form

Problem from the video: Find the missing side. Express your answer in simplest radical form.

The missing side is a leg, so subtract the known leg squared from the hypotenuse squared.

\(10^2+b^2=14^2\)

\(100+b^2=196\)

\(b^2=96\)

\(b=\sqrt{96}\)

Factor out the largest perfect square:

\(b=\sqrt{16\cdot6}\)

\(b=\sqrt{16}\cdot\sqrt6\)

\(b=4\sqrt6\)

✔ Check Your Answer

\(10^2+(4\sqrt6)^2=14^2\)

\(100+16(6)=196\)

\(100+96=196\)

\(196=196\)

Answer: \(\boxed{4\sqrt6\text{ meters}}\)

Example 2: Hypotenuse in Simplest Radical Form

Problem from the video: Find the missing side. Express your answer in simplest radical form.

The missing side is the hypotenuse, so add the squares of the two legs.

\(c^2=4^2+6^2\)

\(c^2=16+36\)

\(c^2=52\)

\(c=\sqrt{52}\)

Factor out the largest perfect square:

\(c=\sqrt{4\cdot13}\)

\(c=\sqrt4\cdot\sqrt{13}\)

\(c=2\sqrt{13}\)

✔ Check Your Answer

\(4^2+6^2=(2\sqrt{13})^2\)

\(16+36=4(13)\)

\(52=52\)

Answer: \(\boxed{2\sqrt{13}\text{ cm}}\)

Video Connection

These examples demonstrate two common situations. In the first problem, a leg is missing, so subtraction is used before simplifying the radical. In the second problem, the hypotenuse is missing, so addition is used before simplifying the radical. In both cases, checking the answer confirms the solution is correct.

🧠 Math Vocabulary

• Pythagorean Theorem: A rule for right triangles that states \(a^2+b^2=c^2\).
• Hypotenuse: The longest side of a right triangle located across from the right angle.
• Leg: One of the two sides that form the right angle.
• Radical: An expression containing a square root.
• Simplest Radical Form: A radical expression with no perfect square factors remaining inside the radical.
• Perfect Square: A number that is the square of an integer, such as 16 or 25.
• Square Root: A value that, when multiplied by itself, equals the original number.
• Verify: To check that an answer satisfies the original equation.

💡 Main Idea

In this lesson, students use the Pythagorean theorem to find missing side lengths and express answers in simplest radical form by factoring out perfect square factors.

📚 What You Should Already Know

Students should know how to identify the hypotenuse, square numbers, find square roots, and recognize perfect squares such as 4, 9, 16, 25, 36, and 49.

🚀 What Comes Next

After learning how to simplify radical answers, students can solve more advanced Pythagorean theorem problems involving coordinate planes, special right triangles, composite figures, and distance formula applications.