Final answer:
To find the length of altitude BP in triangle ABC, use the Pythagorean theorem to find the length of AC, which is the hypotenuse of triangle ABC. Then, substitute the value of x back into the equation AP = x to find the length of altitude BP.
Step-by-step explanation:
To find the length of altitude BP in triangle ABC, we need to determine the relationship between AP and PC. Since BP and CQ are altitudes, we know that AP and PC are perpendicular to BC and AB, respectively. This means that triangle APC is a right triangle, and we can use the Pythagorean theorem to find the length of AC.
Using the Pythagorean theorem, we have:
AC^2 = AP^2 + PC^2
Substituting the given values, we have:
AC^2 = x^2 + (2x - 6)^2
Simplifying and expanding the equation, we have:
AC^2 = x^2 + 4x^2 - 24x + 36
AC^2 = 5x^2 - 24x + 36
Since AC is the hypotenuse of triangle ABC, it is also equal to 3 times the radius of the circumcircle of triangle ABC. Therefore, we have:
3x = sqrt(5x^2 - 24x + 36)
Solving for x, we find:
x = sqrt(5x^2 - 24x + 36)/3
To find the length of altitude BP, we substitute the value of x back into the equation AP = x. Therefore, the length of altitude BP is x.