Step-by-step explanation:
we are using the right triangle altitude theorem. it means that in a right-angled triangle the geometric mean of the two segments of the Hypotenuse (the baseline opposite of the 90° angle) equals the altitude.
in other words
h² = p×q
h = height
p, q are the 2 segments of the Hypotenuse.
so, in our case we have
100 = x(x + 21) = x² + 21x
that gives us the following quadratic equation
x² + 21x - 100 = 0
the general solution to a quadratic equating is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = 21
c = -100
so,
x = (-21 ± sqrt(21² - 4×1×-100))/(2×1) =
= (-21 ± sqrt(441 + 400))/2 = (-21 ± sqrt(841))/2 =
= (-21 ± 29)/2
x1 = (-21 + 29)/2 = 8/2 = 4
x2 = (-21 - 29)/2 = -50/2 = -25
x2 would give us negative lengths for the triangle, which does not make sense.
so, x = 4 is our solution.
that makes the segments of the Hypotenuse 4 and (4+21) = 25 units long (the whole baseline is then 4+25 = 29).
