Pythagorean Triples Multiples Word Problems with Solutions for Grade 8

📚 Understanding Pythagorean Triples

A Pythagorean triple consists of three positive integers $a$, $b$, and $c$, such that $a^2 + b^2 = c^2$. These triples represent the side lengths of a right-angled triangle, where $c$ is the length of the hypotenuse (the side opposite the right angle).

📜 History and Background

The concept of Pythagorean triples dates back to ancient times. The Babylonians, as early as 1800 BC, possessed knowledge of these triples. The most famous triple, (3, 4, 5), was known and used in construction and surveying long before Pythagoras. While Pythagoras is credited with proving the general relationship ($a^2 + b^2 = c^2$) for all right-angled triangles, the understanding and application of specific triples predates him.

📐 Key Principles of Pythagorean Triples

• ➕ Definition: A set of three positive integers $a$, $b$, and $c$ that satisfy the Pythagorean theorem: $a^2 + b^2 = c^2$.
• ➗ Multiples: If $(a, b, c)$ is a Pythagorean triple, then $(ka, kb, kc)$ is also a Pythagorean triple for any positive integer $k$.
• 🎯 Primitive Triples: A Pythagorean triple is primitive if $a$, $b$, and $c$ are coprime (i.e., their greatest common divisor is 1).
• 📝 Generating Triples: Pythagorean triples can be generated using the formula: $a = m^2 - n^2$, $b = 2mn$, $c = m^2 + n^2$, where $m$ and $n$ are positive integers and $m > n$.

🌍 Real-World Examples and Word Problems

Pythagorean triples show up in many real-world scenarios. Here are a few examples structured as word problems:

1. 🪜 Ladder Problem

   A 15-foot ladder leans against a wall. The base of the ladder is 9 feet from the wall. How high up the wall does the ladder reach?

   Solution:

   Let $a$ be the height the ladder reaches on the wall. We have a right triangle with hypotenuse 15 and base 9. Using the Pythagorean theorem:

   $a^2 + 9^2 = 15^2$

   $a^2 + 81 = 225$

   $a^2 = 144$

   $a = 12$

   The ladder reaches 12 feet up the wall.

2. ⛵ Sailing Problem

   A sailboat sails 8 miles east and then 6 miles north. How far is the sailboat from its starting point?

   Solution:

   The eastward and northward movements form a right triangle. The distance from the starting point is the hypotenuse.

   Let $c$ be the distance from the starting point.

   $6^2 + 8^2 = c^2$

   $36 + 64 = c^2$$100 = c^2$

   $c = 10$

   The sailboat is 10 miles from its starting point.

3. 📺 Television Problem

   A rectangular television screen has a width of 24 inches and a height of 18 inches. What is the length of the diagonal of the screen?

   Solution:

   The diagonal of the screen forms the hypotenuse of a right triangle.

   Let $c$ be the length of the diagonal.

   $18^2 + 24^2 = c^2$

   $324 + 576 = c^2$

   $900 = c^2$

   $c = 30$

   The length of the diagonal is 30 inches.

4. ⚾ Baseball Diamond Problem

   A baseball diamond is a square with sides of 90 feet. How far is it from home plate to second base?

   Solution:

   The distance from home plate to second base is the diagonal of the square, forming a right triangle.

   Let $c$ be the distance from home plate to second base.

   $90^2 + 90^2 = c^2$

   $8100 + 8100 = c^2$

   $16200 = c^2$

   $c = \sqrt{16200} = 90\sqrt{2} \approx 127.28$

   The distance from home plate to second base is approximately 127.28 feet.

5. 🌲 Tree Problem

   A tree is supported by a wire that is anchored to the ground 6 feet from the base of the tree. The wire is 10 feet long. How tall is the tree?

   Solution:

   The tree, ground, and wire form a right triangle.

   Let $a$ be the height of the tree.

   $a^2 + 6^2 = 10^2$

   $a^2 + 36 = 100$

   $a^2 = 64$

   $a = 8$

   The tree is 8 feet tall.

6. 🗺️ Navigation Problem

   A ship sails 45 miles due south and then 24 miles due west. How far is the ship from its original position?

   Solution:

   The southward and westward movements form a right triangle.

   Let $c$ be the distance from the original position.

   $45^2 + 24^2 = c^2$

   $2026 + 576 = c^2$

   $2601 = c^2$

   $c = 51$

   The ship is 51 miles from its original position.

7. 🚩 Flagpole Problem

   A 26-foot flagpole is supported by a wire that is attached to the top of the flagpole and anchored to the ground 10 feet from the base of the flagpole. What is the length of the wire?

   Solution:

   The flagpole, ground, and wire form a right triangle.

   Let $c$ be the length of the wire.

   $26^2 + 10^2 = c^2$

   $676 + 100 = c^2$

   $776 = c^2$

   $c = \sqrt{776} \approx 27.86$

   The length of the wire is approximately 27.86 feet.

💡 Conclusion

Pythagorean triples are fundamental in geometry and have practical applications in various fields. Understanding these triples and their multiples helps in solving a wide range of problems related to right-angled triangles.