unit pythagorean theorem homework 2 answer key

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02-02-2025

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This answer key provides solutions for a hypothetical Unit 2 Pythagorean Theorem homework assignment. Since I don't have access to your specific assignment, I'll offer example problems and solutions covering common concepts, allowing you to apply these methods to your own homework. Remember to always show your work!

Understanding the Pythagorean Theorem

Before diving into the answers, let's revisit the core concept: The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (called legs or cathetus). This is expressed as:

a² + b² = c²

Where:

• a and b are the lengths of the legs.
• c is the length of the hypotenuse.

Example Problems and Solutions

Here are some example problems that typically appear in a Pythagorean Theorem unit, along with detailed solutions:

Problem 1: Finding the Hypotenuse

A right-angled triangle has legs of length 3 cm and 4 cm. Find the length of the hypotenuse.

Solution:

1. Identify the knowns: a = 3 cm, b = 4 cm
2. Apply the Pythagorean Theorem: 3² + 4² = c²
3. Calculate: 9 + 16 = c² => 25 = c²
4. Solve for c: c = √25 = 5 cm

Therefore, the length of the hypotenuse is 5 cm.

Problem 2: Finding a Leg

A right-angled triangle has a hypotenuse of length 10 m and one leg of length 6 m. Find the length of the other leg.

Solution:

1. Identify the knowns: c = 10 m, a = 6 m (or b = 6m; it doesn't matter which leg you assign 'a' or 'b' to)
2. Apply the Pythagorean Theorem: 6² + b² = 10² (or a² + 6² = 10²)
3. Calculate: 36 + b² = 100
4. Isolate b²: b² = 100 - 36 = 64
5. Solve for b: b = √64 = 8 m

Therefore, the length of the other leg is 8 m.

Problem 3: Word Problem Application

A ladder is leaning against a wall. The base of the ladder is 5 feet from the wall, and the top of the ladder reaches 12 feet up the wall. How long is the ladder?

Solution:

This is a real-world application of the Pythagorean Theorem. The ladder, wall, and ground form a right-angled triangle.

1. Identify the knowns: a = 5 feet, b = 12 feet
2. Apply the Pythagorean Theorem: 5² + 12² = c²
3. Calculate: 25 + 144 = c² => 169 = c²
4. Solve for c: c = √169 = 13 feet

Therefore, the ladder is 13 feet long.

Problem 4: Dealing with Decimals

A right triangle has legs of length 2.5 cm and 6.0 cm. Find the length of the hypotenuse to one decimal place.

Solution:

1. Identify the knowns: a = 2.5 cm, b = 6.0 cm
2. Apply the Pythagorean Theorem: 2.5² + 6.0² = c²
3. Calculate: 6.25 + 36 = c² => 42.25 = c²
4. Solve for c: c = √42.25 ≈ 6.5 cm

Therefore, the length of the hypotenuse is approximately 6.5 cm.

Beyond the Basics: Further Exploration

These examples cover fundamental applications. Your homework might also include problems involving:

• Converse of the Pythagorean Theorem: Determining if a triangle is a right-angled triangle given the lengths of its sides.
• Three-dimensional problems: Applying the Pythagorean Theorem multiple times to solve problems involving three-dimensional shapes.
• Trigonometric ratios: Relating the angles and side lengths of a right-angled triangle.

Remember to carefully read each problem, draw a diagram if necessary, and show all your steps. If you're still struggling with a particular problem after reviewing this guide and your class notes, seek help from your teacher or a tutor. They can provide personalized assistance and address any specific questions you have.