100 POINTS, MARK BRAINLIEST, 1 QUESTION Answer with a Square-root symbol. (Just say it like 5 root 2)

Step-by-step explanation:

There are three right-angled triangles and two of them have x in their dimensions:

The largest one with one side at 48+x and another at 10

The smallest one with one side at x and another at 10

Those 2 right-angled triangles are similar because of AAA - both have a right-angle and share a common angle.

So their sides are in the same ratio:

x / 10 = 10 / (48+x)

x*(48+x) = 10*10

x^2 + 48x - 100 = 0

(x+50)*(x-2) = 0

x = -50 or 2

As length cannot be negative, x = 2

Step-by-step explanation:

let the missing side in the upper right triangle be y

x^2 + y^2 = 10^2

let the missing side in the lower left triangle be z

48^2 + y^2 = z^2

subtracting the two eqns

48^2 - x^2 = z^2 - 10^2

z^2 = 48^2 + 10^2 - x^2

for the overall big triangle

z^2 + 10^2 = (48+x)^2

substituting

48^2 + 10^2 - x^2 + 10^2 = 48^2 + 96x + x^2

2x^2 + 96x - 200 = 0

x^2 + 48x - 100 =0

(x+50)(x-2) = 0

x=-50 or 2

rejecting the -ve soln

x=2

yes

yes yes yes