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. Read more

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

2. 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.

3. 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: $$ BC^2 = 14^2 + 22^2 $$

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

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

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

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