Using The Pythagorean Theorem For 2 Connected Right Triangles

🔑 Key Concepts

• The Pythagorean theorem works only with right triangles.
• The hypotenuse is always the longest side and is across from the right angle.
• The legs are the two sides that form the right angle.
• When a diagram has two connected right triangles, solve one triangle first, then use that result in the next triangle.
• Simplest radical form keeps exact square root answers instead of rounding too early.

✏️ Worked Examples

Example 1: Find the Shared Diagonal

Problem from the video: The larger right triangle has a hypotenuse of 8 cm and one leg of 5 cm. Find the diagonal shared by the two triangles.

Use the Pythagorean theorem.

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

The hypotenuse is \(8\), so it goes in place of \(c\).

\(5^2+d^2=8^2\)

\(25+d^2=64\)

\(d^2=39\)

\(d=\sqrt{39}\)

Answer: The shared diagonal is \(\boxed{\sqrt{39}}\) cm.

Example 2: Use the Diagonal to Find x

Problem from the video: The smaller right triangle has legs of 3 cm and \(x\) cm. Its hypotenuse is \(\sqrt{39}\) cm. Find \(x\).

Use the Pythagorean theorem again.

\(3^2+x^2=(\sqrt{39})^2\)

\(9+x^2=39\)

\(x^2=30\)

\(x=\sqrt{30}\)

The radical \(\sqrt{30}\) cannot be simplified because 30 has no perfect-square factor greater than 1.

Answer: \(\boxed{x=\sqrt{30}}\) cm, which is about \(5.5\) cm.

Video Problem Diagram

This is the diagram referenced in the worked examples. The key is to notice that the diagonal inside the figure belongs to both right triangles.

🧠 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.
• Adjacent: Next to or touching. Adjacent sides share a common endpoint.
• Radical: A square root expression, such as \(\sqrt{30}\).
• Simplest Radical Form: A radical expression with no perfect-square factor left inside the square root.
• Approximation: A decimal estimate of an exact value, such as \(\sqrt{30}\approx5.5\).

💡 Main Idea

In this lesson, students use the Pythagorean theorem to solve a multi-step right triangle problem. The first right triangle is used to find a missing diagonal, and that diagonal becomes the hypotenuse of a second right triangle used to find \(x\).

📚 What You Should Already Know

Students should know how to square numbers, find square roots, identify the hypotenuse of a right triangle, and solve one-step and two-step equations.

🚀 What Comes Next

After solving missing side problems, students can apply the Pythagorean theorem to word problems, coordinate plane distances, diagonal lengths, and three-dimensional geometry problems.

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