Three cities form a right triangle on a map. Cities B and C are the furthest apart. If cities A and B are 14 miles apart and cities A and C are 22 miles apart, how far apart are ci... Three cities form a right triangle on a map. Cities B and C are the furthest apart. If cities A and B are 14 miles apart and cities A and C are 22 miles apart, how far apart are cities B and C? Round to the nearest hundredth.

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Understand the Problem

The question is asking how far apart cities B and C are, given that three cities form a right triangle, with specified distances between cities A and B, and A and C. The problem can be solved using the Pythagorean theorem.

Answer

The distance between cities B and C is approximately $26.08$ miles.
Answer for screen readers

The distance between cities B and C is approximately $26.08$ miles.

Steps to Solve

  1. Identify the lengths of the triangle sides The sides of the right triangle are given as follows:
  • The distance between cities A and B is 14 miles.
  • The distance between cities A and C is 22 miles.

Let:

  • $AB = 14$ miles
  • $AC = 22$ miles
  1. Use the Pythagorean theorem In a right triangle, the Pythagorean theorem states that: $$ c^2 = a^2 + b^2 $$ Where $c$ is the length of the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the lengths of the other two sides. Here, cities B and C are the furthest apart, indicating that $BC$ is the hypotenuse.

  2. Substitute the known values In this case:

  • $a = AB = 14$
  • $b = AC = 22$

Now substitute those into the Pythagorean theorem: $$ BC^2 = AB^2 + AC^2 $$ This translates to: $$ BC^2 = 14^2 + 22^2 $$

  1. Calculate the squares Calculate $14^2$ and $22^2$: $$ 14^2 = 196 $$ $$ 22^2 = 484 $$

  2. Add the squared lengths Now add the results: $$ BC^2 = 196 + 484 = 680 $$

  3. Take the square root to find distance BC Now take the square root to find $BC$: $$ BC = \sqrt{680} $$ Using a calculator, we find: $$ BC \approx 26.08 $$

  4. Round to the nearest hundredth The final step is to round the answer to the nearest hundredth, which gives: $$ BC \approx 26.08 \text{ miles} $$

The distance between cities B and C is approximately $26.08$ miles.

More Information

The Pythagorean theorem is a fundamental concept in geometry that helps determine the relationship between the sides of a right triangle. It can be applied to various real-world situations, including navigation and architecture.

Tips

  • Forgetting to square the side lengths before adding them together.
  • Confusing which side is the hypotenuse, especially if not clearly stated. Ensure understanding of right triangle properties to avoid this.

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