Using the Pythagorean Theorem in Composite Figures

🔑 Key Concepts

• The Pythagorean theorem can be used inside larger geometric figures.
• Sometimes you must identify a hidden right triangle before solving for the missing length.
• In a trapezoid, drawing a vertical height can create a right triangle.
• In a square, the diagonal forms the hypotenuse of a right triangle.
• Multi-step geometry problems often require using one result to solve the next part of the problem.

✏️ Worked Examples

Example 1: Hidden Right Triangle in a Trapezoid

Problem from the video: What is the value of \(x\)?

The top base is \(9\) inches and the full bottom base is \(17\) inches. The difference creates the horizontal leg of the hidden right triangle.

\(17-9=8\)

The height is \(6\) inches, so the hidden right triangle has legs \(6\) and \(8\).

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

\(6^2+8^2=x^2\)

\(36+64=x^2\)

\(100=x^2\)

\(\sqrt{100}=x\)

Answer: \(x=\boxed{10\text{ in.}}\)

Example 2: Find the Diagonal of a Square

Problem from the video: Determine the length of the square’s diagonal to the nearest centimeter.

First, use the right triangle on top to find the side length of the square.

\(3^2+4^2=c^2\)

\(9+16=c^2\)

\(25=c^2\)

\(5=c\)

So each side of the square is \(5\) cm.

Now use the square’s diagonal as the hypotenuse of a right triangle with legs \(5\) and \(5\).

\(5^2+5^2=d^2\)

\(25+25=d^2\)

\(50=d^2\)

\(d=\sqrt{50}\)

\(\sqrt{50}\approx7.1\)

Answer: The diagonal is about \(\boxed{7\text{ cm}}\).

Video Connection

Both examples show that the Pythagorean theorem is not limited to simple right triangle diagrams. In the trapezoid, you create a hidden right triangle by comparing the base lengths. In the composite shape, you use one right triangle to find the side of a square, then use another right triangle to find the square’s diagonal.

🧠 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.
• Trapezoid: A quadrilateral with at least one pair of parallel sides.
• Composite Figure: A figure made from two or more simpler shapes.
• Diagonal: A segment that connects two non-adjacent vertices of a polygon.
• Nearest Centimeter: A rounded measurement to the closest whole centimeter.

💡 Main Idea

In this lesson, students use the Pythagorean theorem to solve multi-step geometry problems involving a trapezoid, a square, and a composite figure. The key is to find the hidden right triangle and use it to determine the missing length.

📚 What You Should Already Know

Students should know how to identify right triangles, find the hypotenuse, subtract side lengths, square numbers, find square roots, and round decimal answers to the nearest whole number.

🚀 What Comes Next

After solving these problems, students can apply the Pythagorean theorem to coordinate plane distance, diagonal lengths, word problems, three-dimensional figures, and more advanced composite geometry problems.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code for this video.