Final answer:
Using the Pythagorean theorem, we deduce that the other two sides of the right triangle are 12 cm for the base and 5 cm for the height when the hypotenuse is 13 cm.
Step-by-step explanation:
To find the lengths of the other two sides of a right triangle where the height is 7 cm less than its base and the hypotenuse is 13 cm, we can use the Pythagorean theorem. Let's denote the base as 'b' and the height as 'b - 7'. Applying the theorem we have: b² + (b - 7)² = 13².
Let's solve the equation step-by-step:
- b² + (b - 7)² = 169. Expanding (b - 7)² gives b² + b² - 14b + 49.
- Combining like terms yields 2b² - 14b + 49 = 169.
- Subtracting 169 from both sides gives 2b² - 14b - 120 = 0.
- Dividing by 2, we get b² - 7b - 60 = 0, which is a quadratic equation.
- Factoring the quadratic equation, we find (b - 12)(b + 5) = 0.
- Setting each factor equal to zero gives us b = 12 (since b = -5 is not a viable length for a triangle side).
- Therefore, the base is 12 cm and the height is 12 - 7 = 5 cm.
So, the sides of the triangle measure 12 cm and 5 cm.