In triangle ABC, AB is 8 cm, AC is 14 cm. Find the length of BC.

Steps to Solve

1. Identify the sides of the triangle

   In triangle ABC, we have:

   • Side AB (height) = 8 cm
   • Side AC (base) = 14 cm
   • Side BC is the hypotenuse, which we need to find.

2. Apply the Pythagorean theorem

   The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (BC) is equal to the sum of the squares of the lengths of the other two sides (AB and AC).

   This can be expressed as: $$ BC^2 = AB^2 + AC^2 $$

3. Substitute the values into the equation

   Now we will substitute the known values: $$ BC^2 = 8^2 + 14^2 $$

4. Calculate the squares

   Calculate the squares of AB and AC:

   • $8^2 = 64$
   • $14^2 = 196$

   Substitute these values back: $$ BC^2 = 64 + 196 $$

5. Add the results

   Add the results together: $$ BC^2 = 260 $$

6. Find BC by taking the square root

   Finally, take the square root of both sides to find the length of BC: $$ BC = \sqrt{260} $$

   Simplifying this gives: $$ BC \approx 16.12 \text{ cm} $$