Final answer:
To find the value of x in the right triangle with a 13 cm hypotenuse and the height being 7 cm less than the base, we apply the Pythagorean theorem and solve the quadratic equation, resulting in x being 12 cm.
Step-by-step explanation:
The student asks about finding the value of x in a right triangle where the altitude (height) is 7 cm less than the base, and the hypotenuse is 13 cm using the Pythagorean theorem. To find the base x, we use the relationship that the altitude y is x - 7 cm.
Written as an equation with the altitude as x - 7 cm, this becomes:
x² + (x - 7)² = 13²
Then we expand and simplify the equation:
x² + x² - 14x + 49 = 169
This simplifies to:
2x² - 14x - 120 = 0
Dividing the entire equation by 2 yields:
x² - 7x - 60 = 0
Solving this quadratic equation, we find that x = 12 cm (ignoring the negative solution as a side length cannot be negative). Therefore, the base of the right triangle is 12 cm.