Solved Examples: Grade 8 Multi-Step Real-World Pythagorean Theorem Problems.

📚 Quick Study Guide

• 📐 Pythagorean Theorem: The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is: $a^2 + b^2 = c^2$, where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.
• 💡 Identifying the Hypotenuse: The hypotenuse is always the longest side of the right-angled triangle and is opposite the right angle.
• 📝 Real-World Applications: The Pythagorean Theorem is used in various real-world scenarios such as navigation, construction, and determining distances.
• ➕ Multi-Step Problems: These problems often require you to use the Pythagorean Theorem in combination with other mathematical concepts like area, perimeter, or volume.
• 🧭 Problem-Solving Steps:
  1. Read the problem carefully and identify what you need to find.
  2. Draw a diagram if possible to visualize the problem.
  3. Apply the Pythagorean Theorem or related formulas.
  4. Solve for the unknown variable.
  5. Check your answer to make sure it makes sense in the context of the problem.

Practice Quiz

1. A ladder is leaning against a wall. The base of the ladder is 5 feet away from the wall, and the ladder reaches a height of 12 feet on the wall. How long is the ladder?
   1. 11 feet
   2. 13 feet
   3. 15 feet
   4. 17 feet
2. A rectangular garden is 8 meters long and 6 meters wide. What is the length of the diagonal of the garden?
   1. 9 meters
   2. 10 meters
   3. 12 meters
   4. 14 meters
3. A baseball diamond is a square with sides of 90 feet. What is the distance from home plate to second base?
   1. 90 feet
   2. $90\sqrt{2}$ feet
   3. 120 feet
   4. 180 feet
4. A ship sails 10 km east and then 24 km north. How far is the ship from its starting point?
   1. 22 km
   2. 26 km
   3. 34 km
   4. 36 km
5. A right triangle has legs of length 7 cm and 24 cm. What is the length of the hypotenuse?
   1. 23 cm
   2. 25 cm
   3. 31 cm
   4. 33 cm
6. A telephone pole is 40 feet tall. A wire is stretched from the top of the pole to a point on the ground 30 feet away from the base of the pole. How long is the wire?
   1. 35 feet
   2. 45 feet
   3. 50 feet
   4. 70 feet
7. A park has a rectangular playground that is 24 meters wide and 32 meters long. They plan to divide it diagonally with a fence. How long will the fence be?
   1. 36 meters
   2. 40 meters
   3. 48 meters
   4. 56 meters