8th grade pythagorean theorem worksheet

Lemon

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03-02-2025

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This worksheet is designed to help 8th-grade students master the Pythagorean Theorem and its applications. We'll cover the basics, explore practical examples, and tackle some challenging problems to solidify your understanding. Remember, the Pythagorean Theorem is a fundamental concept in geometry, with applications far beyond the classroom!

Understanding the Pythagorean Theorem

The Pythagorean Theorem describes the relationship between the sides of a right-angled triangle. A right-angled triangle is a triangle with one 90-degree angle (a right angle). The theorem states:

a² + b² = c²

Where:

• a and b are the lengths of the two shorter sides (legs or cathetus) of the right-angled triangle.
• c is the length of the longest side (hypotenuse), which is opposite the right angle.

This formula means that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Example:

Imagine a right-angled triangle with sides a = 3 cm and b = 4 cm. To find the length of the hypotenuse (c), we apply the theorem:

3² + 4² = c²

9 + 16 = c²

25 = c²

c = √25 = 5 cm

Therefore, the hypotenuse is 5 cm long.

Practice Problems: Finding the Hypotenuse

Use the Pythagorean Theorem (a² + b² = c²) to solve for the hypotenuse (c) in the following right-angled triangles:

1. a = 6 cm, b = 8 cm, c = ?
2. a = 5 cm, b = 12 cm, c = ?
3. a = 9 cm, b = 12 cm, c = ?
4. a = 7 cm, b = 24 cm, c = ?
5. a = 2 cm, b = 2 cm, c = ?

Practice Problems: Finding a Leg

Now let's try finding the length of one of the shorter sides (legs) when you know the hypotenuse and one leg. Remember to rearrange the formula to solve for 'a' or 'b': a² = c² - b² or b² = c² - a²

1. c = 13 cm, b = 5 cm, a = ?
2. c = 10 cm, a = 6 cm, b = ?
3. c = 15 cm, b = 9 cm, a = ?
4. c = 17 cm, a = 8 cm, b = ?
5. c = 5 cm, a = 3 cm, b = ?

Real-World Applications

The Pythagorean Theorem isn't just a classroom exercise; it has numerous real-world applications:

• Construction: Builders use it to ensure corners are square and to calculate diagonal measurements.
• Navigation: It's used in GPS systems and surveying to determine distances.
• Computer Graphics: It's fundamental to creating 2D and 3D images and animations.
• Sports: Calculating distances in various sports like baseball or basketball.

Challenge Problems

1. A ladder 10 meters long is leaning against a wall. The base of the ladder is 6 meters from the wall. How high up the wall does the ladder reach?

2. A rectangular field is 20 meters long and 15 meters wide. What is the length of the diagonal path across the field?

3. A right-angled triangle has a hypotenuse of 25 cm and one leg of 15 cm. What is the area of the triangle? (Remember: Area of a triangle = ½ * base * height)

Remember to show your work for each problem! Understanding the process is as important as getting the right answer. Good luck, and happy problem-solving!