Pythagorean Theorem Word Problems

🔑 Key Concepts

• Pythagorean theorem word problems often describe a right triangle without showing one at first.
• Words like north, south, east, and west usually describe horizontal or vertical legs.
• Words like directly across, diagonal, ramp, ladder, or shortest path often describe the hypotenuse.
• When two perpendicular distances are given, use \(a^2+b^2=c^2\) to find the direct distance.
• Always answer the question being asked, which may require one extra step after finding the hypotenuse.

✏️ Worked Examples

Example 1: Walking Around a Lake

Problem from the video: Malik walks 6 miles south and 8 miles east to get from point A to point B. If he could go directly across the lake, how many miles shorter would the distance be?

The words south and east tell us the two walking paths form the legs of a right triangle.

The phrase directly across tells us to find the diagonal distance, which is the hypotenuse.

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

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

\(36+64=c^2\)

\(100=c^2\)

\(\sqrt{100}=c\)

\(10=c\)

The direct distance across the lake is \(10\) miles.

Walking around the lake is \(6+8=14\) miles.

\(14-10=4\)

Answer: The direct route would be \(\boxed{4\text{ miles}}\) shorter.

Example 2: Wheelchair Ramp

Problem from the video: A wheelchair ramp rises 3 feet off the ground and extends 20 feet along the ground. How long will the ramp be? Round to the nearest tenth of a foot.

The phrase rises 3 feet gives the vertical leg.

The phrase extends 20 feet along the ground gives the horizontal leg.

The ramp itself is the slanted distance, so it is the hypotenuse.

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

\(3^2+20^2=c^2\)

\(9+400=c^2\)

\(409=c^2\)

\(\sqrt{409}=c\)

\(c\approx20.2237\)

Answer: The ramp is about \(\boxed{20.2\text{ ft}}\) long.

Video Connection

Both word problems require students to translate the situation into a right triangle. In the lake problem, south and east are the legs and the direct path is the hypotenuse. In the ramp problem, the rise and run are the legs and the ramp is 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.
• Direct Distance: The straight-line distance between two points.
• Horizontal Distance: A side-to-side distance, often described with east or west.
• Vertical Distance: An up-and-down distance, often described with north, south, rise, or height.
• Nearest Tenth: A rounded value with one digit after the decimal point.

💡 Main Idea

In this lesson, students solve Pythagorean theorem word problems by identifying the legs and hypotenuse in real-world situations involving a lake and a wheelchair ramp.

📚 What You Should Already Know

Students should know how to identify the hypotenuse, square numbers, find square roots, round decimals, and use the Pythagorean theorem to find a missing side of a right triangle.

🚀 What Comes Next

After solving these word problems, students can apply the Pythagorean theorem to maps, coordinate plane distance, diagonal measurements, construction problems, and other real-world geometry situations.

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