Explanation:
imagine this triangle to be a trigonometric triangle in a circle.
the angle is at the triangle vertex at the center of the circle.
13×sqrt(2) is then the cosine of 45° multiplied by the Hypotenuse (= radius of the circle) x.
13×sqrt(2) = cos(45)×x
x = 13×sqrt(2)/cos(45) = 26
y = sin(45)×x = 13×sqrt(2)×sin(45)/cos(45) =
= 13×sqrt(2)×tan(45) = 13×sqrt(2)×1 = 18.38477631...
as expected, a right-angled triangle with 45° base angle must be an isoceles triangle (both legs are equally long).
y = 13×sqrt(2)
x = 26
so, if you are not supposed to use trigonometric functions yet, we can wrap it up in the opposite way :
because it is a right-angled triangle with a base angle of 45° (which makes also the other base angle 45°), it is an isoceles triangle, and therefore
y = 13×sqrt(2)
x we get then via Pythagoras :
x² = (13×sqrt(2))² + (13×sqrt(2))² = 169×2 + 169×2 =
= 4×169 = 676
x = sqrt(676) = 26