Applying the Pythagorean Theorem

🔑 Key Concepts

• The Pythagorean theorem can be used to find a missing side of a right triangle.
• When the hypotenuse and one leg are given, subtract the leg squared from the hypotenuse squared.
• Once the missing height or base is found, it can be used in another formula, such as area of a triangle.
• Radical side lengths can still be used in the Pythagorean theorem by squaring the radicals.
• Always identify the hypotenuse first because it is the side across from the right angle.

✏️ Worked Examples

Example 1: Find Area After Finding a Missing Side

Problem from the video: What is the area of the triangle?

The triangle has a base of \(12\) cm and a hypotenuse of \(13\) cm. To find the area, we first need the missing height.

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

\(12^2+b^2=13^2\)

\(144+b^2=169\)

\(b^2=25\)

\(b=5\)

Now use the area formula for a triangle.

\(A=\frac{bh}{2}\)

\(A=\frac{12(5)}{2}\)

\(A=\frac{60}{2}\)

Answer: The area is \(\boxed{30\text{ cm}^2}\).

Example 2: Find a Missing Side With Radicals

Problem from the video: Find the length of the missing side.

The two given sides are legs, so the missing side is the hypotenuse.

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

\((\sqrt{3})^2+(\sqrt{6})^2=c^2\)

Squaring a square root gives the number inside the radical.

\(3+6=c^2\)

\(9=c^2\)

\(\sqrt{9}=\sqrt{c^2}\)

\(3=c\)

Answer: The missing side is \(\boxed{3}\).

Video Connection

Both problems use the same big idea: identify the right triangle, decide which side is the hypotenuse, and apply \(a^2+b^2=c^2\). In the first example, the missing side is needed before finding area. In the second example, radical side lengths are squared to find the hypotenuse.

🧠 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.
• Area of a Triangle: The amount of space inside a triangle, found using \(A=\frac{bh}{2}\).
• Base: The side of a triangle used as the bottom or reference side when finding area.
• Height: The perpendicular distance from the base to the opposite vertex.
• Radical: A square root expression, such as \(\sqrt{6}\).

💡 Main Idea

In this lesson, students use the Pythagorean theorem in two different situations: finding a missing height before calculating the area of a triangle, and finding a missing side when the given side lengths include radicals.

📚 What You Should Already Know

Students should know how to identify the hypotenuse, square numbers, square simple radicals, solve basic equations, and find the area of a triangle using base and height.

🚀 What Comes Next

After practicing these examples, students can apply the Pythagorean theorem to word problems, coordinate plane distance, diagonal lengths, and multi-step geometry problems involving area or surface area.

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